Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Area of ellipse which is not in standard form By graphing device i understand that $x^2+xy+y^2=1$ is ellipse. By some geometry i find area of above ellipse which comes out $\pi$ (is it right?), but it was easy case. Is there any quick method or standard formula to calculate it or we have to convert it into standard for... | If the conic $ Q(x,y) = ax^2 + by^2 + c + 2hxy + 2fy + 2gx$ is an ellipse, then if we translate the orgin of our coordinate system to the centre of $ Q$ to get,
$$ {R(X,Y) = Q(X + u, Y + v)} = aX^2 + b Y^2 + 2hXY + c^\prime$$
where $(u, v)$ is the centre of the conic given. We need to find $(u, v)$ and $c^\prime$ (pra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Suppose $p$ is an odd prime and there exists some integer $x$ such that $x^2 \equiv -1 \pmod{p}$. Prove that $p \equiv 1 \pmod{4}$. Suppose $p$ is an odd prime and there exists some integer $x$ such that $x^2 \equiv -1 \pmod{p}$. Prove that $p \equiv 1 \pmod{4}$. Use Fermat's Theorem.
| Let's call the order of $x$ to be the smallest positive integer power, $k$ so that $x^k \equiv 1 \mod p$. We'll denote the order of $x$ as $|x|$.... (assuming some a power exists; which it won't if $x$ is not relatively prime to $p$.)
Clearly $x^{k|x| + j; 0 \le j < |x|} = (x^{|x|})^k*x^j \equiv 1^k*x^j\equiv x^j \mod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Minimum and maximum value related to the sides of the quadrilateral.
If $a$, $b$, $c$ and $d$ are the sides of the quadrilateral then find the minimum value of
$$\frac{a^2+b^2+c^2}{d^2}.$$
I have tied by the inequality $a+b+c>d$, but it doesn't work.
| The maximum does not exist: $a=b=c\rightarrow+\infty$.
The minimum does not exist.
Indeed, by C-S $$\frac{a^2+b^2+c^2}{d^2}=\frac{(1+1+1)(a^2+b^2+c^2)}{3d^2}\geq\frac{(a+b+c)^2}{3d^2}>\frac{d^2}{3d^2}=\frac{1}{3}.$$
Since for $a=b=c\rightarrow\frac{d}{3}$ we have $\frac{a^2+b^2+c^2}{d^2}\rightarrow\frac{1}{3}$, we obta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove the inequality $a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$ for positive $a, b, c$ I faced problem proving this inequality for positive $a$, $b$, $c$:
$a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$
I tried to simplify it and I got that:
$bc^3 + a^3 c + a b^3 \geq a b^2 ... | We need to prove that
$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq a+b+c,$$
which is true by Rearrangement because
$(a^2,b^2,c^2)$ and $\left(\frac{1}{a},\frac{1}{b},\frac{1}{c}\right)$ are opposite ordered,
which gives
$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\geq\frac{a^2}{a}+\frac{b^2}{b}+\frac{c^2}{c}=a+b+c.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why the remainders of the division of a number by 2 results in the number in binary? For example 14, if you divide be 2 it equals 7 and the remainder = 0, then 7/2 = 3 with r = 1 then 3/2 = 1 and r = 1 then 1/2 = 0 with r = 1. If we take the r's and put them togheter 1110 is 14 in binary. I don't understand the intuiti... | This is because of Horner's scheme for evaluating polynomials:
\begin{align}&a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}\dots+a_1x+a_0\\
={}(\dots(((&a_nx+a_{n-1})x+a_{n-2})x+\dots+a_1)x+a_0.
\end{align}
The successive divisions by $2$ yield
\begin{align}
14&=7\cdot 2+0\\&=(3\cdot2+1)\cdot 2+0=3\cdot2^2+1\cdot 2+0\\
&=\color{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2460284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Computing $\int \frac{1}{\sqrt{25y^2-30y-7}} \,dy$ $$\int \frac{1}{\sqrt{25y^2-30y-7}} \, dy$$
I start by completing the square: so that $(5y-3)^2-16=25y^2-30y-7$.
Then I let $u=5y-3$, and then let $ u=4 \sec \theta$.
This leads to the equation becoming $\frac{1}{4\tan\theta}$ which is the same as $\frac{1}{4}\cot \the... | HINT: set $$\sqrt{25y^2-30y-7}=5y+t$$ then you will get
$$y=-\frac{7+t^2}{10t+30}$$
$$dy=-1/10\,{\frac { \left( t+7 \right) \left( t-1 \right) }{ \left( t+3
\right) ^{2}}}dt
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2460655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Solving $x_{n} - 3x_{n-1} = -8$ with $n\geq 1$ and $x_0 = 2$ I tried two methods which gave different answers:
Method 1:
$$x_{n} - 3x_{n-1} = -8 \\ x_n = 3(3x_{n-2} - 8) - 8 \\ = 3^2 x_{n-2} -8 ( 1+3) \\ = 3^3 x_{n-3} - 8(1+3+3^2) \\ = 3^n x_{0} - 8(1+3+3^2 + \ldots + 3^{n-1}) \\ = 2\times 3^n - 8\left(\frac{3^n - 1}... | Safety way
Set
$v_n =x_n-a$ such that $v_n$ satisfies $$ v_{n+1} =3v_n\Longleftrightarrow x_{n+1}-a =3(x_n-a)$\Longleftrightarrow 3x_{n}-8-a =3(x_n-a)\Longleftrightarrow a=4$$
Then
$$v_n =3^nv_0\Longleftrightarrow x_n-4 = 3^n(x_0-4)\Longleftrightarrow \color{red}{x_n = -2\cdot3^n +4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2461496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that there exist infinitely many primitive Pythagorean triples $x, y, z$ whose even member $x$ is a perfect square. Prove that there exist infinitely many primitive Pythagorean triples $x, y, z$ whose even member $x$ is a perfect square. [Hint: consider the triple $4n^2, n^4-4, n^4+4$, where $n$ is an arbitraty o... | If $n$ is odd, $\gcd (n^4-4, n^4+4)=\gcd (n^4-4,8)=1$ by the Euclidean algorithm.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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minimum value of a two variable problem Suppose that $x$ and $y$ are in $(−2, 2)$ and $xy = −1$. The minimum value of
$\frac4{4-x^2}+\frac9{9-y^2}$ is ?
So I have tried to manipulate this equation but what I got is a complex equation with square roots in it...
| $x, y \in (−2, 2)$ and $xy = −1$.
Let $3x+2y=k$. Then $9x^2-12+4y^2=k^2$. That is, $9x^2+4y^2=k^2+12$.
\begin{align}
f(x,y)
&= \frac4{(4-x^2)}+\frac9{(9-y^2)} \\
&= \dfrac{72 - (9x^2+4y^2)}{36-(9x^2+4y^2)+1} \\
&= \dfrac{72 - (k^2+12)}{36-(k^2+12)+1} \\
&= \dfrac{60 - k^2}{25-k^2} \\
&= 1 + \dfrac{35}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove this inequality $1\le \sum _{cyc}\frac{a}{1+bc}\le \frac{3\sqrt{3}}{4}$ Let $a,b,c>0$ such that $a^2+b^2+c^2=1$. Prove that
$$1\le\dfrac{a}{1+bc}+\dfrac{b}{1+ca}+\dfrac{c}{1+ab}\le\dfrac{3\sqrt{3}}{4}$$
*) $LHS\le \frac{3\sqrt{3}}{4} \Rightarrow LHS^2\le \frac{27}{16}$
$\Leftrightarrow \left(1+1+1\right)\left(\s... | Firstly, the left inequality.
By C-S and AM-GM we obtain:
$$\sum_{cyc}\frac{a}{1+bc}=\sum_{cyc}\frac{a^2}{a+abc}\geq\frac{(a+b+c)^2}{\sum\limits_{cyc}(a+abc)}\geq$$
$$\geq\frac{(a+b+c)^2}{a+b+c+\frac{(a+b+c)(ab+ac+bc)}{3}}=\frac{3(a+b+c)}{3+ab+ac+bc}.$$
Thus, it remains to prove that
$$3(a+b+c)\geq3+ab+ac+bc$$ or
$$9(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Arithmetic of complex numbers
If $z = \cos x + i\sin x$ , show that
$$\frac{2}{1+z} =1-i\tan\left(\frac{x}{2}\right),$$
I get up to:
$$\frac{1+\cos x -i\sin x}{1+\cos x}.$$
| Note that
\begin{align*}\frac{2}{1+z}&=\frac{2(1+\overline{z})}{|1+z|^2}=\frac{2(1+\cos x - i\sin x)}{(1+\cos x)^2 + (\sin x)^2}\\
&=\frac{2(1+\cos x - i\sin x)}{2(1+\cos x)}=1-i\cdot\frac{\sin x}{1+\cos x}
\end{align*}
and recall the tangent half-angle formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Solving Recurrence Relation of $T(n)=4T(n-2)+2$ Question
Solve Recurrence Relation of $T(n)=4T(n-2)+2$
Base case-: $T(1)=1,T(2)=2$
My Approach/solution
$$T(n)=4T(n-2)+2$$
$$T(n-2)=4T(n-4)+2 \tag{1}$$
$$T(n-4)=4T(n-6)+2 \tag{2}$$
Using $(1)$ and $(2)$ in my equation
$$\begin{align*}
T(n)&=4\cdot (4T(n-4)+2)+2\\
&=4^{2... | Using generating functions technique we have
$$f(x)=\sum\limits_{n=0}T(n)\cdot x^n=T_0+1\cdot x+2\cdot x^2+\sum\limits_{n=3}T(n)\cdot x^n=\\
T_0+1\cdot x+2\cdot x^2+\sum\limits_{n=3}\left(4T(n-2)+2\right)\cdot x^n=\\
T_0+4\sum\limits_{n=3}T(n-2)\cdot x^n + x + 2\cdot x^2 +\sum\limits_{n=3}2\cdot x^n=\\
T_0+4x^2\sum\li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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prove that $\lim_{(x,y)\to (0,0)}\frac{x^2y-xy^2}{x^2+y^1}=0$ I need prove that
$$\lim_{(x,y)\to (0,0)}\frac{x^2y-xy^2}{x^2+y^2}=0$$
Can I use it?
if $\sqrt{x^2+y^2}< \delta$ then $|\frac{x^2y-xy^2}{x^2+y^2}|<\epsilon$
$$\left |\frac{x^2y-xy^2}{x^2+y^2} \right |=\frac{x^2 \left |y \right | -\left |x \right |y^2}{x^2+y... | If the question is correct:
$f(x,y) = \frac {xy^2 - x^2y}{x^2 + y}$
Suppose $y = -x^2 + \zeta$
where $\zeta$ is something very small.
$f(x,y) \approx \frac{x^2}{\zeta}$
There exists an $(x,y)$ in a neighborhood of $(0,0)$ such that $f(x,y)$ is large.
The limit does not exist.
If it is
$f(x,y) = $$\frac {xy^2 - x^2y}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466708",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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"answer_id": 2
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Counting with combinations vs manually Lets say there is an urn containing 10 balls, 5 are red and 5 are black. 3 are drawn without replacement. I am finding the probability that I get 3 red balls.
I understand that I can count this in two different ways:
1.) (5/10)(4/9)(3/8)
2.) 5C3/10C3
and that when I expand and si... |
Find the probability that three red balls are drawn when three balls are drawn without replacement from an urn containing five red and five black balls.
You correctly found the answer in two ways. In the first, you counted ordered selections. In the second, you counted unordered selections. However, this is not th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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minimize $x^4 - 6x^2 y^2 + y^4$ given $x^2 + y^2 \leq 1$ I have a constrained optimization problem. Can we maximize / minimize this function on the unit sphere?
$$ f(x,y,z) = x^4 - 6 x^2 y^2 + y^4 \quad\text{given that}\quad x^2 + y^2 + z^2 = 1$$
One idea could be to use the Cauchy-Schwartz inequality. Since I forg... | A different approach. As you noted, being on or within the unit sphere just implies $x^2+y^2\leq 1$, and we can then ignore $z$.
$$F = x^4 - 6x^2 y^2 + y^4$$
$$F = (x^2-y^2)^2 - 4x^2 y^2$$
$$F = (x^2-y^2+2xy)(x^2-y^2+2xy)$$
Now substitute $x = r \cos\theta, y = r \sin\theta$. We know $0 \leq r\leq 1$.
$$F = (r^2\cos^2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $a+b+c=0$ prove that $ \frac{a^2+b^2+c^2}{2} \times \frac{a^3+b^3+c^3}{3} = \frac{a^5+b^5+c^5}{5} $
If $a+b+c=0$, for $a,b,c \in\mathbb R$, prove
$$ \frac{a^2+b^2+c^2}{2} \times \frac{a^3+b^3+c^3}{3} = \frac{a^5+b^5+c^5}{5} $$
I've tried squaring, cubing, etc. the $a+b+c=0$, but I've just dug myself in.
Is there a... | Let' s try like this: Suppose $a,b,c$ are the roots of the polynomial $x^3-mx-n$.
By Vieta's formula's we have:
$$a^2+b^2+c^2 = (a+b+c)^2-2(ab+bc+ca)= 0-2m = 2m$$
Since $x^3 =mx+n \;\;\;\;(*)$ we have:
$$x^5 = x^2(mx+n) = mx^3+nx^2= m^2x+mn+nx^2$$
so
$$a^5+b^5+c^5 = m^2(a+b+c)+3mn+n(a^2+b^2+c^2) =3mn+n(a^2+b^2+c^2)=5... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How many bit strings of length $n$ contain exactly $k$ blocks of "$10$"? How many bit strings of length $n$ contain exactly $k$ blocks of "$10$"?
My attempt: Let $F(n, k)$ be the number of bit strings of length $n$ that contain exactly $k$ blocks of $10.$ Note that for $k \neq0, $ $F(0, k) = F(1, k)= 0.$
Consider a bit... | Essentially this amounts to finding the places where the string switches from a run of $1$'s to a run of $0$'s. Note that there are $n-1$ places where such a switch may occur (between each pair of numbers). In any event, there must be $k$ places where the string switches from $1$ to $0$:
*
*If the string starts and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question on uniqueness of projections in linear algebra Suppose A and B are projection matrices and suppose that for some vector x in the column space of B, BAx=x. Can I say that Ax=x?
Since x is in the column space of B, I know that Bx=x it projects itself in its own column space. I guess my question is, is x the only... | Let $A = \begin{pmatrix}0&1\\0&1\end{pmatrix},$ $B = \begin{pmatrix}0&0\\0&1\end{pmatrix},$ and $x = \begin{pmatrix}0\\1\end{pmatrix}.$ Then $A^2 = A$ and $B^2 = B,$ so $A$ and $B$ are indeed projection matrices. On the other hand,
$$
\begin{pmatrix}0&0\\0&1\end{pmatrix}\begin{pmatrix}0&1\\0&1\end{pmatrix}\begin{pmatr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2473648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $\frac {2 - \csc^2 A}{\csc^2 A\space + \space2\cot A} \equiv \frac {\sin A \space -\space \cos A} {\sin A \space+\space \cos A}$ So I have this rather simple trigonometric identity that, for the life of me, I cannot solve. I have worked on it for about 2 hours and still can't get it.
Show that
$$\frac {2 - \c... | =$\frac{2 sin^2(A)-1}{1+2sin(A)cos(A)}$
=$\frac{2sin^2(A)-(cos^2(A)+sin^2(A))}{(cos^2(A)+sin^2(A))+2sin(A)cos(A)}$
=$\frac{sin^2(A)-cos^2(A)}{(sin(A)+cos(A))^2}$
=$\frac{(sin(A)+cos(A))(cos(A)+sin(A))}{(sin(A)+cos(A))^2}$
=$\frac{sin(A)-cos(A)}{(sin(A)+cos(A))}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2474872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Inverse of Quadratics with Horizontal Transformations This might sound like a beginner question for a lot of the community, I apologize, but, I really need help understanding it. I was trying to find the inverse of the quadratic $ f(x)= (\frac 12x +2)^2 +4$. Using WolframAlpha, I was able to find that the inverse funct... | $y=(\frac 12x +2)^2 +4$
$x=(\frac 12y +2)^2 +4$
$x-4=(\frac 12y +2)^2$
Everything is right up to here.
$\sqrt{x^2} = |x|$, so you need a $\pm$ in front of the square root, because you need to account for both positive and negative values of $y$ for a given $x$ while solving for the inverse.
$\pm \sqrt{x-4} =\frac 12y +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2477815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Conjugate of Quaternion
The conjugate of $$\alpha=\left[ {\begin{array}{cc}
a+bi & c+di \\
-c+di & a-bi \\
\end{array} } \right]$$ is$$\overline{\alpha}=\left[ {\begin{array}{cc}
a-bi & -c-di \\
c-di & a+bi \\
\end{array} } \right]$$
The norm of $\alpha$ is $a^2+b^2+c^2+d^2$ and is written $\lvert\lver... | You do not need to bother with matrix multiplication being commutative. Once you have figured out the matrix product, multiplication of the complex numbers within each component of that product is still commutative. What doesn't always commute is multiplication of one matrix by another, which you are already done wit... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find $\lim_{n\to\infty} \frac{1^4+2^4+\dots+n^4}{1^4+2^4+\dots+n^4+(n+1)^4}$ I am just trying to calculate
$$\lim_{n\to\infty} \frac{1^4+2^4+\dots+n^4}{1^4+2^4+\dots+n^4+(n+1)^4}.$$
To do this I apply formula for sum of fourth powers of $n$ number. My result: $$\lim_{n\to\infty}\frac{1^4+2^4+\dots+n^4}{1^4+2^4+\dots+n... | From this result: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
Then $\dfrac{1^4+2^4+...+n^4+(n+1)^4}{1^4+2^4+...+n^4}=1+\dfrac{(n+1)^4}{O(n^5)}=1+O(\frac 1n)\to 1$ and so is the inverse of that.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2480039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
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Recurrence induction finding a constant $b$ such that $f(n) \leq bn$
Pretend induction is just a weird way our teacher uses induction when it comes to finding constant. Same as induction
Base Case:
let $n = 1$
$f(n) = 6(1) = 6$ and $bn = b$. Therefore we need $b \geq 6$ (*)
let $n = 2$
$f(n) = 6(2) = 6$ and $bn = 2b$.... | I think the problem is that you are using $\frac{n+3}{4}\leq\frac{2n}4$ which isn't strong enough (if the problem had $\lfloor 2n/4\rfloor$ instead of $\lceil n/4\rceil$ no such $b$ would exist).
Instead, keep $\frac{n+3}{4}$. That means that $$f(n)\leq\bigg(\frac{35}{36}b+5+\frac{3b}{2n}\bigg)n.$$
We $n$ to be quite ... | {
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"answer_id": 0
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Basis for Linear Transformation with Matrix Multiplication Let $V$ be the vector space of $ 2 \times 2 $ real matrices.
Find a basis for the kernel of the linear transformation $T :V \rightarrow V$ given by $T(A)= XA $.
$X$, being the matrix below.
\begin{bmatrix}1&1\\1&1\end{bmatrix}
The matrix $X$ isn't invertible, s... | You are looking for the set of all vectors $x \in V$ such that $T(x) = 0$, which is the definition of the null space (kernel). So then you want to look for the elements in $V$ such that
$$T(X) = 0 $$
First we observe the transformation
$$T\left(\begin{pmatrix} a&b\\c&d\end{pmatrix}\right) = \begin{pmatrix} 1&1\\1&1\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Inequality with cosine and sine Let $A:=f^2+g^2$, where $f,g$ are functions of $x$ such that
$$f'=(c-1)(f\cos(x)\sin(x)+g\sin^2(x)),$$
$$g'=-(c-1)(f\cos^2(x)+g\cos(x)\sin(x)),$$
for some constant $c$.
(Note: $f'\cos(x)+g'\sin(x)=0$)
How do I show that $A'\leq4|c-1|A?$
I see that
\begin{align}
A'&=2ff'+2gg'\\
&=2(c-1)... | Cauchy-Schwartz' inequality tells you that
$$
|a f + b g| \le \sqrt{a^2 + b^2}\sqrt{f^2+ g^2}
$$
Starting from your result, we obtain
$$
A^\prime\le |A^\prime|\le 2 |c-1|1.\sqrt{f^2+g^2}1.\sqrt{f^2+g^2} = 2 |c-1|A
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2483931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Prove the intersection between $z^2=x^2+y^2$ and $x+y+2z=2$ is an ellipse. I want to prove that the intersection between the cone $z^2=x^2+y^2$ and the plane $x+y+2z=2$ is an elispe in this plane.
My work:
I suppose to prove it I have to see that the equation $x+y+2\sqrt{x^2+y^2}=2$ can be rewritten as $\frac{x^2}{a}... | You cannot put it in this form, because the ellipse is not always centered at the origin, or having its major axis on either the $y$ or $x$ axes.
$$\frac{x+y-2}{2}=z$$
$$\left(\frac{x+y-2}{2}\right)^2=x^2+y^2$$
$$x^2+xy-2x+xy+y^2-2y-2x-2y+4=4x^2+4y^2$$
$$3x^2+3y^2-2xy+4x+4y=4$$
This conic is an ellipse. You can use or... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to evaluate $\int_{0}^{+\infty} \cos(x^2)\cos(x)dx$ How do I evaluate $\int_{0}^{+\infty} \cos(x^2)\cos(x)dx$?
I don't know what to do. Should I use a contour integration?
| Let $I$ be the desired integral. Then, by the product-to-sum trigonometric identity,
\begin{align*}
2I&=\int_{0}^{+\infty} \cos (x-x^2)\,dx+\int_{0}^{+\infty} \cos (x^2+x)\,dx\\
&=\int_{0}^{+\infty} \cos ((x-1/2)^2-1/4)\,dx+\int_{0}^{+\infty} \cos ((x+1/2)^2-1/4)\,dx\\
&=\cos(1/4)\int_{0}^{+\infty} \cos ((x-1/2)^2)\,dx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Show that series is convergent and find its sum $$\sum\nolimits\arccos\frac{n(n+1)+\sqrt{(n+1)(n+2)(3n+1)(3n+4)}}{(2n+1)(2n+3)}, n\in\mathbb{N}^{*}$$
I need help proving that this series is convergent and calculating its sum.
What I've done so far:
$$\lim_{n\to\infty}\arccos\frac{n(n+1)+\sqrt{(n+1)(n+2)(3n+1)(3n+4)}}{(... | To know the answer, first you must know some results:
the propose (1) is sum of angle of cosine:
$$\arccos\left(x_1\right)-\arccos\left(x_2\right)=\arccos\left(x_1\cdot x_2+\sqrt{\left(1-x_1^2\right)\left(1-x_2^2\right)}\ \right)\quad, x_1<x_2$$
hence
$$ \arccos\left(x_1\right)-\arccos\left(x_2\right)=\arccos\left(x_1\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2484894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Tossing a biased coin We toss a biased coin $k$ times with the probability of tossing a head being $1/m$. We know that in the first two attempts there was at least one tail. What the probability that we tossed exactly $n$ heads?
Well, the tosses are independent so getting at least one tail in the first two attempts is ... | The probability that we tossed exactly n heads, given there was exactly 1 tail in the first two tries, is $Q_{1}={{k-2}\choose{n-1}}(\frac{m-1}{m})^{n-1}(\frac{1}{m})^{k-n-1}$, and the probability that we tossed exactly n heads, given there was exactly 2 tails in the first two tries, is $Q_{2}={{k-2}\choose{n}}(\frac{m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve system $\cos^2x+\cos^2y+\cos^2z=1,$ $\cos x+\cos y+\cos z=1,$ $x+y+z=\pi$ I want to solve the system:
$$\cos^2(x)+\cos^2(y)+\cos^2(z)=1,$$
$$\cos(x)+\cos(y)+\cos(z)=1,$$
$$x+y+z=\pi.$$
I tried to prove that only one of cosines can be not a zero, but I just prove that one or three cosines can not be zero.
I get,... | Since $z=\pi-x-y$, you have $\cos z=\cos(x+y)$; then
\begin{align}
\cos^2x&+\cos^2y+\cos^2z
\\[6px]
&=\cos^2x+\cos^2y+\cos^2x\cos^2y-2\cos x\cos y\sin x\sin y+\sin^2x\sin^2y
\\[6px]
&=\cos^2x+\cos^2y+\cos^2x\cos^2y-2\cos x\cos y\sin x\sin y
\\
&\qquad+1-\cos^2x-\cos^2y+\cos^2x\cos^2y\\[6px]
&=1+2\cos x\cos y(\cos x\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Number of solutions of $\sin (2x)+\cos (2x)+\sin x+\cos x=1$ Find Number of solutions of $\sin (2x)+\cos (2x)+\sin x+\cos x=1$ in $\left [0 \:\: 2\pi\right]$
The equation can be written as:
$$\sin (2x)+1-2 \sin^2x+\sin x+\cos x=1$$ $\implies$
$$\sin x+\cos x=2\sin^2 x-2 \sin x\cos x$$ $\implies$
$$\sin x+\cos x=2\sin ... | I think your reasoning with graphs is not so right because if so,
why not to draw the graph of $f(x)=\sin2x+\cos2x+\sin{x}+\cos{x}-1$?
By the Claude's hint from your equation
$$\frac{1+\tan{x}}{1-\tan{x}}=-2\sin{x}$$ after substitution $\tan\frac{x}{2}=t$ we obtain
$$\frac{1+\frac{2t}{1-t^2}}{1-\frac{2t}{1-t^2}}=-2\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2486920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Proof of an inequality by induction Prove using induction that
$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \cdots+ \frac{1}{n^2} \le 2-\frac{1}{n}$$
for all positive whole numbers $n$.
I began by showing that it is true for $n=1$
I then assumed that it is true for $n=p$
$$\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{... | I think the following induction a bit of better.
$$\sum_{k=1}^n\frac{1}{k^2}=1+\sum_{k=2}^n\frac{1}{k^2}<1+\sum_{k=2}^n\frac{1}{k(k-1)}=1+\sum_{k=2}^n\left(\frac{1}{k-1}-\frac{1}{k}\right)=1+1-\frac{1}{n}=2-\frac{1}{n}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2488093",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Solving the integral :$\int_0^{\frac{\pi}{2}}\frac{\sin x}{1+\sqrt{\sin2 x}}dx$ Solving the integral :
$$\int_0^{\frac{\pi}{2}}\frac{\sin x}{1+\sqrt{\sin 2x}}dx=?$$
My try:
$$\int_0^{\frac{\pi}{2}}\frac{\sin x}{1+\sqrt{\sin 2x}}\cdot \frac{1-\sqrt{\sin 2x}}{1-\sqrt{\sin 2x}}dx$$
$$\int_0^{\frac{\pi}{2}}\frac{\sin x(1-... | The given integral equals
$$ \frac{1}{2}\int_{0}^{\pi}\frac{\sin\tfrac{x}{2}}{1+\sqrt{\sin x}}\,dx =\frac{1}{2\sqrt{2}}\int_{0}^{\pi}\frac{\sqrt{1-\cos x}}{1+\sqrt{\sin x}}\\=\frac{1}{2\sqrt{2}}\int_{0}^{\pi/2}\frac{\sqrt{1-\cos x}+\sqrt{1+\cos x}}{1+\sqrt{\sin x}}\,dx\\
=\frac{1}{2}\int_{0}^{\pi/2}\frac{\sqrt{1+\sin x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2492880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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On the summation $\sum \limits_{n=1}^{\infty} \arctan \left ( \frac{1}{n^3+n^2+n+1} \right )$ Here is a problem that I ran into. I seriously doubt if there is a closed form but you never know.
Evaluate the series
$$\mathcal{S} = \sum_{n=1}^\infty \arctan \left ( \frac 1 {n^3+n^2+n+1} \right) $$
I searched in vain to at... | We have
$$S = \sum_{n=1}^{\infty}\arctan\left(\frac{1}{n^3 + n^2 + n + 1}\right) \\ = \Im\left(\ln\left(\prod_{n=1}^{\infty} \left(1 + \frac{i}{n^3+n^2+n+1}\right)\right)\right) \\ = \Im\left(\ln\left(\prod_{n=1}^{\infty} \frac{n^3+n^2+n+1+i}{n^3+n^2+n+1}\right)\right) \\ = \Im\left(\ln\left(\frac{\pi\operatorname{csch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2495407",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
What would the determinant of the following matrix be? " If $$\det\begin{pmatrix}a&1&d\\ b&1&e\\ c&1&f\end{pmatrix}=4$$
and $$\det
\begin{pmatrix}a&1&d\\ b&2&e\\ c&3&f\end{pmatrix}=3$$
What is $$\det \begin{pmatrix}a&-1&d\\ b&-3&e\\ c&-5&f\end{pmatrix}?$$"
This does not seem to fit into any of the regular changes in th... | $\det \begin{pmatrix}a&-1&d\\ b&-3&e\\ c&-5&f\end{pmatrix}=\det\begin{pmatrix}a&1-2\times1&d\\ b&1-2\times 2&e\\ c&1-2\times 3&f\end{pmatrix}=\det \begin{pmatrix}a&1&d\\ b&1&e\\ c&1&f\end{pmatrix}-2\det \begin{pmatrix}a&1&d\\ b&2&e\\ c&3&f\end{pmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
How to find the number of solutions for $x_1+2x_2+5x_3=10$? This is a question taken from Discrete mathematics by Kenneth Rosen:
Find the number of ways to make change for \$100 using \$10, \$20 and \$50 bills.
My approach:
Let number of \$10 notes be $x_1$.
Let number of \$20 notes be $x_2$.
Let number of \$50 notes b... |
first variant
When looking for non-negative integral solutions $x_1,x_2,x_3$ we notice that the possible solutions of $x_3$ in
\begin{align*}
x_1+2x_2+5x_3=10\tag{1}
\end{align*}
are $x_3\in\{0,1,2\}$ since $0\leq 5x_3\leq 10$.
Setting these three values for $x_3$ we consider instead of (1) the three equations
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2496774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to solve $ T(n) = 2 T(\frac{n}{8}) + \Theta(\sqrt[3]{n}) $ I'm tasked to give the best possible asymptotic bounds for the following recurrence:
$$ T(n) = 2 T\left( \frac n 8 \right) + \Theta(\sqrt[3]{n}) $$
I got the following using the iterative method:
\begin{align} T(n) & = 2 T \left( \frac n 8 \right) + \Theta... | The strictest proof goes like this. Following the definition of $\Theta(\sqrt[3]{n})$, there exists two constants $C_1$ and $C_2$, such that for all sufficiently large $n\geq n_0$, where $n_0$ is a constant (may depend on $C_1,C_2$)
$$C_1\sqrt[3]{n}\leq T(n)-2T(\frac{n}{8})\leq C_2\sqrt[3]{n}.$$
Then multiply everybody... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2497398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Linearization of a Differential Equation Can someone please help me to linearize this system, which is given by the differential equation shown in the picture below.
All variables are expressed as deviations from initial values
(0, for all variable states (x, x)).
My task is to find the model described as the one sho... |
$$m \ddot x+2c(x^2-1) \dot x+kx=0. \tag{1}$$
We define $y:=\dot x$ and plugging this into (1) + some algebra yields
\begin{align} m \dot y+2c(x^2-1) y+kx=0 \\ \Longleftrightarrow \dot y=-\frac{2c}{m}(x^2-1)y-\frac{k}{m}x. \end{align}
To get back into the notation of your picture, we set $\mathbf{X}:=(x,y)^T$ and get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2500148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find the exact value of $\cos\frac{2\pi}{5}$ by solving equation There's this concept in the topic of complex numbers which I don't really understand much (and was devastated upon realising it'll appear in the topic often) - trigonometry!
I'm super lost to be honest.
We are asked to express $\cos3\theta$ and $\cos2\the... | Here I use the formulas
$$\cos(x+y) = \cos x \cos y - \sin x \sin y$$
$$\sin(x+y) = \sin x \cos y + \cos x \sin y.$$
So $$\cos 2\theta = \cos(\theta+\theta) = \cos^2 \theta -\sin^2 \theta = 2\cos^2 \theta - 1.$$
Similarly
\begin{align}
\cos 3\theta = \cos (2\theta+\theta) & = \cos 2\theta\cos \theta-\sin 2\theta \sin \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2505537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
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Does $x^2 \equiv 3$ (mod $q$) (where $q$ is an odd prime) have infinite solutions? Not sure how to prove/disprove this. One thought I had for proving this was doing an indirect proof, assuming there are only finitely many solutions $x_1,x_2,...,x_n$ and perhaps:
1) constructing a new solution using these solutions
or ... | There is an ambiguity in the question. I address the version of the question "For how many primes $q$ does $x^2 \cong 3 \pmod{q}$ have a solution?" (I ignore the other interpretation because either we recognize that $\mathbb{Z} / q \mathbb{Z}$ is a finite set of residue classes, so there are only finitely many classe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2508318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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How to solve $a^3 + 39 ab^2 - 18 = 0$, $3a^2 b + 13 b^3 - 5 = 0$ In this answer, the user @123 has claimed that by solving the system
$$\begin{cases}a^3 + 39 ab^2 - 18 = 0 \\ 3a^2 b + 13 b^3 - 5 = 0\end{cases}$$
we give $ a = \dfrac 32$ and $ b = \dfrac12$. Could anyone explain for me that how one can solve such a syst... | Here is one approach. Note from the second equation you have
$$
3a^2 = 5/b - 13b^2
$$
which we can substitute into the first to get
$$
18 = a\left(a^2+39b^2\right)
= a\left(\frac{5}{3b} - \frac{13b^2}{3}+39b^2\right)
$$
so
$$
3a^2 = \left(\frac{18\cdot 3}{\frac{5}{3b} - \frac{13b^2}{3}+39b^2} \right)^2
$$
which impl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2511064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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How to find the first same-proportion rectangle(s) whose both sides are also whole numbers? Let's elaborate the issue with an example:
I have an arbitrary rectangle. Let's choose the rectangle to be $5.6 \times 4.2$. The width-to-height ratio is $\frac 4 3$. The rectangle is in between the two whole-numbered rectangles... | General algorithm
Let's say we start from a rectangle with side lengths $w$ and $h$ (in your case $w=5.6$ and $h=4.2$). Now you try to express $w/h$ as a rational number $p/q$ with $\gcd(p,q)=1$. In your case we have
$$\frac{5.6}{4.2}=\frac{56}{42}=\frac{\color{red}2\cdot 2\cdot 2\cdot \color{blue}7}{\color{red}2\cdot ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2513697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Finding the convolution of two chi squared variables? If I have variables $X_i^2$ that are chi squared distributed with $1$ degree of freedom: $$f_{X_i^2}=\frac 1 {\sqrt{2\pi}}\frac 1 {\sqrt x}e^{-\frac x 2}$$
Then I want to derive the distribution of $Z^2=X_1^2+X_2^2$ where these $X's$ are i.i.d.
We do this by convol... | Let $u=\sqrt{x} \implies \frac{du}{dx}=\frac1{2\sqrt{x}} \implies 2\, du = \frac1{\sqrt{x}}\, dx$. Also $u^2=x$ implies $y-x=y-u^2$. So your integral is
$$\int_0^{\sqrt{y}}\frac{dx}{\sqrt{x(y-x)}}=\int_0^{\sqrt{y}}\frac{2}{\sqrt{y-x}}\cdot \frac1{2\sqrt{x}}\, dx=\int_0^{\sqrt{y}}\frac{2}{\sqrt{y-u^2}}\, du$$ as $0\le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2516718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Maximum area of a triangle defined by a point $P$ inside another triangle (Try using ceva's theorem) Let $P$ be a point inside an acute triangle $\triangle ABC$ and let $D$, $E$, and
$F$ be the points of intersection of the lines $AP$, $BP$ and $CP$ with sides $BC$, $CA$ and $AB$, respectively. Determine $P$ so that th... | Let $\frac{AF}{FB}=x$, $\frac{BD}{DC}=y$ and $\frac{CE}{EA}=z$.
Thus, by Cheva's theorem $xyz=1$ and
$$\frac{S_{\Delta AFE}}{S_{\Delta ABC}}=\frac{AF\cdot AE}{AB\cdot AC}=\frac{x}{(1+x)(1+z)},$$
$$\frac{S_{\Delta BFD}}{S_{\Delta ABC}}=\frac{BD\cdot BF}{BA\cdot BC}=\frac{y}{(1+y)(1+x)},$$
$$\frac{S_{\Delta CDE}}{S_{\D... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2519928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to simplify solutions for $y'' + 4y = 2 \tan x$? The original equation is $$y'' + 4y = 2 \tan x$$
What I did so far:
$$\lambda^2+4 = 0$$
$$\lambda_1 = -2i \quad \lambda_2 = 2i$$
$$y(x) = C_1\cos2x+C_2\sin2x$$
Write down the system:
$$\begin{cases} C_1'\cos2x + C_2'\sin2x = 0 \\
2C'_2\cos2 x - 2C_1'\sin2x = 2\tan x... | There is an error in your calculation of $\Delta$.
$$ \Delta = \begin{vmatrix}\cos(2x) &\sin(2x) \\ -2\sin(2x) &2\cos(2x)\end{vmatrix} = 2\cos^2(2x)+2\sin^2(2x) = 2$$
$$ C'_1 = \frac{\Delta_2}{\Delta} = -2\sin^2 x$$
$$ C_1 = -2\int sin^2 x \; dx = \frac{1}{2}\sin(2x) -x$$
$$ C'_2 = \frac{\Delta_3}{\Delta} = -\cos(2x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2520293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Finding the roots of simultaneous equations like these
I am unable to find the answer to this question. It would also be helpful if you post what areas of mathematics this relates to.
| Expand $(a+b+c)^3$, giving
$$a^3+b^3+c^3+3a^2b+3ab^2+3b^2c+3bc^3+3c^2b+3cb^2+6abc\\
=3(a^2+b^2+c^2)(a+b+c)-2(a^3+b^3+c^3)+6abc.$$
$$7^3=3\cdot35\cdot7-2\cdot151+6abc$$
$$abc=15.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2524752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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How to solve this partial differential equation ? (PDE) (UPDATED) How can I solve this PDE?
$$ \frac{{∂}^2z}{∂x^2} - \frac{∂^2z}{∂x∂y} - 2\frac{∂^2z}{∂y^2} +6\frac{∂z}{∂x}- 9\frac{∂z}{∂y} +5z = e^{2x +y} + e^{x+y} $$
| The given equation is $(D^2-DD'-2D'^2+6D-9D'+5)z=e^{2x+y}+e^{x+y}$
The auxillary equation is obtained by setting $D=m, D'=1$ since there are no common factors. This gives $m^2-m-2+6m-9+5=0\implies m^2+5m-6=0$. Call the roots $c_1$ and $c_2$. So the homogeneous (complementary) solution is $z_c(x,y)=\phi_1(y+c_1x)+\phi_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2526015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Fourier series of $f(x) = 1$ for $x\in[-\pi,0]$, $f(x) = -1$ for $x\in[0,\pi]$. I am attempting to solve a Fourier series problem where we have the question defined by a piecewise function:
$$f(x) = \begin{cases} 1 & -\pi \leq x \leq 0 \\ -1 & 0 \leq x \leq \pi \end{cases}$$
I can solve it out where I calculate that ... | Just for the Fourier series:
\begin{align}
f(x) &= A_{0} + \sum_{n=1}^{\infty} \left( A_{n} \, \cos(n x) + B_{n} \, \sin(n x) \right) \\
A_{0} &= \frac{1}{2 \pi} \, \int_{-\pi}^{\pi} f(x) \, dx \\
A_{n} &= \frac{1}{\pi} \, \int_{-\pi}^{\pi} f(x) \, \cos(n x) \, dx \\
B_{n} &= \frac{1}{\pi} \, \int_{-\pi}^{\pi} f(x) \, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2528711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Calculation of $\int \sin^2 x\frac{\sin x-\cos x}{\sin x+\cos x}dx$
Calculation of $\displaystyle \int\frac{\sin^2 x\cos x}{
\sin x+\cos x}dx$ and $\displaystyle \int^{\pi}_{0}\frac{1}{1-2a\cos x+a^2}dx, ,0<a<1$
$\bf{Attempt}$ For (a) $\displaystyle \frac{1}{2}\int\frac{\sin^2 x\bigg[(\sin x+\cos x)+(\sin x-\cos x)\b... | For (a),
\begin{eqnarray}
&&\int\frac{\sin^2 x\cos x}{\sin x+\cos x}dx\\
&=&\int \frac{\sin^2x(\cos^2x-\sin x\cos x)}{\cos^2x-\sin^2x}dx\\
&=&\frac14\int \frac{(1-\cos(2x))(1+\cos(2x)-\sin(2x))}{\cos(2x)}dx\\
&=&\frac14\int\frac{1-\cos^2(2x)-(1-\cos(2x))\sin(2x)}{\cos(2x)}dx\\
&=&\frac14\int(\sec(2x)-\cos(2x)-\tan(2x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2529631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$L^1$ norm of derivative of Dirichlet kernel Let $D_n(x)$ denote the $n$-th Dirichlet kernel. We all know that $D_n$ has $L^1$ norm $O(\log n)$ (over a full period). But I'm wondering about the $L^1$ norm of its derivative. Is it $O(n)$ or $O(n \log n)$? Would anyone know a proof? Thanks.
Edit: Here's a proof that it i... | By symmetry, it suffices to look at the interval $[0,\pi]$. From
$$D_n(t) = \frac{\sin \bigl(\bigl(n+\frac{1}{2}\bigr)t\bigr)}{\sin \frac{t}{2}}$$
(with the appropriate interpretation if $t = 0$) we can read off the zeros of $D_n$, they are $z_k = \frac{k\pi}{n+\frac{1}{2}}$ for $1 \leqslant k \leqslant n$. Between $z_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2529928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
How to put a complex number into the form a+ib if it has an exponent So I'm trying to put this into the form a+ib
$(1 + i)^{1000}$
(Hint: Use the polar form of the number).
I know the polar form without the exponent would be
$√2(cos(π/4)+isin(π/4))$
Do i just throw the exponent on at the end? I'm not sure how this hel... | Apropos of the answers from other posters applying DeMoivre's Theorem, the problem can be thought of in terms of the geometrical interpretation of multiplication between complex numbers. When we multiply a number $ \ a + bi \ = \ \sqrt{a^2 + b^2}·cis(\theta) \ \ $ by a second number $ \ c + d \ = \ \sqrt{c^2 + d^2}·ci... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2530152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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I found a correlation between n^x and x! can anyone explain it? This is the example to numbers to the 3rd.
Here is the same thing just with numbers to the 4th
Well I was messing around with numbers I made this discovery, with the correlation between n^x and x!. What is the reason for this? Is it easily explainable?
In ... | The binomial theorem says
$$
(a+b)^x = a^x + xa^{x-1}b + \frac{x(x-1)}2 a^{x-2} b^2 + \frac{x(x-1)(x-2)} 6 a^{x-3} b^3 + \cdots.
$$
The differences in the column to the right of the one that lists values of $a^x$ are
$$
(a+1)^x - a^x = \left( a^x + x a^{x-1} + \frac{x(x-1)} 2a^{x-2} + \cdots \right) - a^x.
$$
Since $a^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2535656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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About factoring and simplifying the following algebra expression... today a problem like the following was sent to me by a friend: $$\dfrac{(a^3+a+10)}{(a^3+3a^2+a-2)}\cdot\dfrac{(a^2-2a+5)}{(a^4-3a^2+1)}$$ find the simplest form of the expression above. The answer as the key suggests turns out to be $a^2-a-1$.
I've ... | Probably there is a typo. The reducible form occurs if the operation is a division, not a multiplication. Accordingly, inverting the numerator and the denominator of the second fraction, the expression becomes
$$\dfrac{(a^3+a+10)}{(a^3+3a^2+a-2)}\cdot\dfrac{( a^4-3a^2+1 )}{( a^2-2a+5 )}$$
and taking into accou... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2536781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find this maximum of the $\frac{\sqrt{3}}{4}x^2+\frac{\sqrt{(9-x^2)(x^2-1)}}{4}$ Let $x\in \mathbb{R}$, find the function maximum of the value
$$f(x)=\dfrac{\sqrt{3}}{4}x^2+\dfrac{\sqrt{(9-x^2)(x^2-1)}}{4}$$
my attemp
$$x^2=5+4\sin{t},t\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$$
then
$$f=\dfrac{5\sqrt{3}}{4}+2\sin{\... | Hint:
$$4f(x)-5\sqrt3=\sqrt{16-(x^2-5)^2}+\sqrt3(x^2-5)$$
Now set $x^2-5=4\cos t,0\le t\le\dfrac\pi2$
We can prove $$a\cos t+b\sin t\le\sqrt{a^2+ b^2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Does there exist a closed form for the sinc function series $\sum_{n=1}^\infty \frac{\sin\sqrt{n^2+1}}{\sqrt{n^2+1}}$? Here I want to get the closed form solution of the following summation
$$
\sum_{n=1}^\infty \frac{\sin\sqrt{n^2+1}}{\sqrt{n^2+1}} \qquad(1)
$$
Or the more general form ($x$ be an arbitrary real number,... | COMMENT to users: Achille-Hui and Random-Variable
How about this ?:
$$\sum _{n=0}^{\infty } \frac{\sin \left(x \sqrt{n^2+a^2}\right)}{\sqrt{n^2+a^2}}=\frac{1}{2} \pi J_0(a x)+\frac{\sin (a x)}{2 a}$$
$$\sum _{n=0}^{\infty } \mathcal{L}_x\left[\frac{\sin \left(x \sqrt{n^2+a^2}\right)}{\sqrt{n^2+a^2}}\right](s)=\frac{1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 4,
"answer_id": 3
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Evaluating $\lim_{n \rightarrow \infty} \left( \frac{2}{\sqrt{n^2 + 4n} - n} \right) ^{B(n+2)}$ I want to solve the following problem:
Find $B \in \mathbb{R}$ such that
$$\lim_{n \rightarrow \infty} \left( \frac{2}{\sqrt{n^2 + 4n} - n} \right) ^{B(n+2)} \in \left] \frac{1}{2}, 2 \right[ \quad.$$
My attempt is as foll... | $$\frac{2}{\sqrt{n^2+4n}-n}=1+\frac{\sqrt{n^2+4n}-n}{2n}$$
$$=1+\frac{a_n}{n}$$ where $a_n\to 1$
It follows that
$$(1+\frac{a_n}{n})^{n+2}\to e.$$ Does this help ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The ± sign in square root As I was doing a SAT question when I came across this question:
$\sqrt {x-a} = x-4$
If $a=2$,what is the solution set of the equation?
Options
*
*{$3,6$}
*{$2$}
*{$3$}
*{$6$} Correct Answer
I evaluated the equation and got $0=(x-3)(x-6)$If you put those number in the equatio... | The standard way to solve equations with square roots is to use this rule:
$$ \sqrt A=B\iff (A=B^2\quad\textbf{and}\quad B\ge 0). $$
So here you obtain
$$\sqrt{x-2}=x-4\iff x-2=x^2-8x+16\;\text{and}\; x\ge 4\iff(x-3)(x-6)=0\;\text{and}\; x\ge 4,$$
which shows there's only one root: $6$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2545302",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
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Number of seven digit numbers without repetition of digits divisible by $3$ Find number of seven digit numbers divisible by $3$ with
$1.$ Repetition
$2.$ Without Repetition
For Part $1.$ The least seven digit number divisible by $3$ is $1000002$ and highest seven digit number is $9999999$
So total is $3000000$
For Part... | $(1)$ Number of possible digit combinations without $0$ digit: $\binom{9}{7}=36 $
$(2)$ Number of possible digit combinations including $0$ digit: $\binom{9}{6}=84 $
So there are $36 \cdot 7!+84 \cdot 6 \cdot 6!=544320$ numbers without digit repetition.
Now we have to find the numbers divisible by $3$.
We can seperate ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to factorize polynomials to the 5th degree? I have the polynomial:
$$2x^5-x^4+10x^3-5x^2+8x-4$$
and I know that the final result is:
$$(2x-1)(x^4+5x^2+4) = (2x-1)(x^2+1)(x^2+4)$$
But how would you do it step by step? I've seen a couple of videos and blogs about it, but they mostly use examples, where their is a com... | The question slightly wrong in my opinion. Please try multiplying the factors or the eqn. and lets see if you will get back the original question.
The correct equation is $P(x): 2x^5-x^4-10x^3+5x^2+8x-4$
Group and Factorize: $(2x^5-x^4) (-10x^3+5x^2) (8x-4) =0$
$x^4(2x-1) -5x^2(2x-1) 4(2x-1)$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2549116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Using Euclidean Algorithm in RSA? it leverages multiplication and subtraction, which humans are fairly good at, to make fractions like 15996751/3870378 reducible. Also useful in scaling equations down to their simplest integer representation in one step, given with extra integers, GCD(C,GCD(A,B)) is equivalent to GCD(A... | Here is the continued fraction way of writing this. It is, of course, equivalent to the "back-substitution" way one often sees. The convergents can bbe written in a single row of fractions, as below.
$$ \gcd( 1680, 71 ) = ??? $$
$$ \frac{ 1680 }{ 71 } = 23 + \frac{ 47 }{ 71 } $$
$$ \frac{ 71 }{ 47 } = 1 + \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2555000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove for any integer $a$ show that $a$ and $a^{4n+1}$ have the same last digit
For any integer $a$, show that $a$ and $a^{4n+1}$ have the same last digit
I know that if $a^{4n+1} \equiv a\pmod{10}$ then $10|a^{4n+1}-a$, so $2|a^{4n+1}-a$ and $5|a^{4n+1}-a$, but I'm not sure where to go from here.
| If $\gcd(a,10) = 1$ then $a^{\phi(10)} = a^4 \equiv 1\mod 10$. So $a^{4n+1} = (a^4)^n*a\equiv 1^na \equiv a \mod 10$.
If $\gcd(a,10) \ne 1$ then $a = 0, 5$ or $2^kb; \gcd(b,10) =1$.
$0^k = 0\equiv 0 \mod 10$ (duh) and $5^k \equiv 5 \mod 10$ (do I really need to show that?)
$6^2 = 36\equiv 6 \mod 10$ so by induction $6... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2557373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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Evaluate the limit $\lim_{x\to \infty} x(16x^4 + x^2+1)^{1/4}-2x^2$ Can someone please check my conclusion to the evaluation of the following limit?
$$\lim_{x\to \infty} x(16x^4 + x^2+1)^{1/4}-2x^2$$
I got that the limit is equal to infinity. If limit is equal to infinity does this mean that limit does not exist?
| Observe that
$$g(x) = 16x^4 + x^2+1 = 16(x^2 + \frac 1{32})^2 + \frac{63}{64}$$
Take the fourth root to get
$$\sqrt[4]{g} \sim 2 \sqrt{x^2 + \frac 1{32}}$$
as $x \to \infty$.
We still have to multiply by $x$ and then subtract $2x^2$. We see that
$$x \cdot 2 \sqrt{x^2 + \frac 1{32}} = 2 \sqrt{x^4 + \frac {x^2}{32}} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2559138",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Help with this determinant I need help to calculate this determinant:
$$\begin{vmatrix}
1^k & 2^k & 3^k & \cdots & n^k \\
2^k & 3^k & 4^k & \cdots & (n+1)^k\\
3^k & 4^k & 5^k & \cdots & (n+2)^k\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
n^k & (n+1)^k & (n+2)^k & \cdots & (2n-1)^k
\end{vmatrix}$$
Where $2\leq n$ and... | Another proof.
This involves some tricks on manipulating polynomials.
$\mathit{Proof}.\blacktriangleleft$
Consider
$$
f(z) =
\begin{vmatrix}
(z+1)^k & (z+2)^k & (z+3)^k & \cdots & (z+n)^k\\
2^k & 3^k & 4^k & \cdots & (n+1)^k\\
3^k & 4^k & 5^k & \cdots & (n+2)^k \\
\vdots & \vdots & \vdots & \ddots & \vdots\\
n^k & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2560282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Can someone explain this proof to me for $2n < 2^n - 1? \,\, n >= 3$ Prove $2n < 2^n - 1;n \geq 3$
Prove for $n = 3$
$2\cdot 3 < 2^6 - 1$
$6 < 7$
Prove for $n \mapsto n + 1$
$$2(n + 1) = 2n + 2$$
$$< 2^n - 1) + 2$$
$$= 2^n + 1$$
$$< 2^n + 2^n - 1$$
$$ = 2^{n + 1} - 1$$
Can someone explain this proof to me? I get... | Your inductive hypothesis is that $2n < 2^n - 1$.
If the inductive hypothesis holds, then by adding 2 to both sides, you find that $$(2n) + 2 < (2^n - 1) + 2$$
On the left hand side, $2n + 2$ is equal to $2(n+1)$.
On the right hand side, $(2^n -1) + 2 = 2^n + 2 - 1 < 2^{n} + 2^{n} - 1 = 2^{n+1} - 1$.
And putting them... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Find the indefinite integral $\int\frac{dx}{(x^2+2x+5)^2}$ I need help with the indefinite integral
\begin{align}
& \int\frac{dx}{(x^2+2x+5)^2} \\[10pt]
= {} & \int\frac{dx}{((x+1)^2+4)^2} = \int\frac{du}{(u^2+4)^2} & & x+1=u,\quad du=dx \\[10pt]
= {} & \frac{1}{16} \int\frac{du}{(\frac{u^2}{4}+1)^2} \\[10pt]
= {} & ... | it is $$x^2+2t+5=(x+1)^2+4$$ we Substitute $$x+1=t$$ then we have $$dx=dt$$
and $$\int\frac{1}{(t^2+4)^2}dt$$ and then we Substitute $$t=2\tan(s)$$ with $$dt=2\sec^2(s)ds$$ and we get $$(t^2+4)^2=16\sec^4(s)$$ and our integral is $$2\int \frac{\cos^2(s)}{16}ds$$ and in the last step note that $$\cos^2(s)=\frac{1}{2}\co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Binomial to the power of five, equality proof I want to find out when the equality $(x+y)^5=x^5+y^5$ for real numbers $y$ and $x$ holds.
Expanding this binomial yields $(x+y)^5=x^5+y^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4$
Factoring the right side of the above equality gives $x^5+y^5+5xy(x^3+2x^2y+2xy^2+y^3)$
If $(x+y)^5=x^5+... | It can be helpful with equations like your last one there to gather terms on one side, and factor. Thus: $$(x+y)^3-xy(x+y)=0\\\implies (x+y)((x+y)^2-xy)=0\\ \implies x+y=0\,\,\,\text{ or }\,\,\, (x+y)^2-xy=0$$
What can you do with that second equation?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solving the Differential equation: $y'=\frac{2}{x}y+x^3$ We have the differential equation $$y'=\frac{2}{x}y+x^3$$ and we know $x \in (0, \infty)$.
My attempt with variation of constants
\begin{align}
\phi(x) &= \exp \left(\int \frac{2}{x} dx \right) \\
&= \exp(2\ln|x|) \\
&= x^2c
\end{align}
and
\begin{align}
\psi(x)... | $$y'=\frac { 2 }{ x } y+x^{ 3 }\\ y'-\frac { 2 }{ x } y=0\\ \frac { dy }{ dx } =\frac { 2y }{ x } \\ \int { \frac { dy }{ y } } =2\int { \frac { dx }{ x } } \\ \ln { y } =2\ln { x } +C\\ \ln { y } =\ln { C{ x }^{ 2 } } \\ y=C{ x }^{ 2 }\\ y=C\left( x \right) { x }^{ 2 }\\ { y }'={ C }'\left( x \right) { x }^{ 2 }+2xC... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 3
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Proving $|f(y)-f(x)|\leq f(|y-x|)$ for $f(x)=-x\log_2(x)$ I need to prove (or disprove) the following claim:
for $0 \leq x \leq y \leq 1$:
$$|f(y)-f(x)|\leq f(y-x)$$
where $f(x) = -x \log_2(x)$ if $x \in (0,1)$ and $f(x)=0$ elsewhere.
I managed to prove that: $f(y)-f(x) \leq f(y-x)$ using Lagrange's mean value theorem... | The claim is false as stated. Indeed, consider the case when $y=1$ and $x=\frac{1}{4}$. Then
$$
f(y) =f(1)= 0 \qquad\text{and}\qquad
f(x) =f\left(\frac{1}{4}\right)=\frac{1}{4}\log(4)
$$
and
$$
f(y-x) = f\left(\frac{3}{4}\right) = \frac{3}{4}\log\left(\frac{4}{3}\right) = \frac{1}{4}\log\left(\frac{4^3}{3^3}\right).
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove: $(\frac{1+i\sqrt{7}}{2})^4+(\frac{1-i\sqrt{7}}{2})^4=1$ $$\left(\frac{1+i\sqrt{7}}{2}\right)^4+\left(\frac{1-i\sqrt{7}}{2}\right)^4=1$$
I tried moving the left exponent to the RHS to then make difference of squares exp. $(x^2)^2$. Didn't get the same on both sides though. Any help?
| Let $(a_n)_n$ the sequence verifying $\begin{cases} a_0=2\\ a_1=1\\a_{n+2}=a_{n+1}-2a_n\end{cases}$
The characteristic equation of this linear recurrence relation is $x^2=x-2$
Whose roots are $\dfrac{1\pm i\sqrt{7}}2$.
Thus $a_n=\alpha r^n+\beta {\bar r}^n$ and given the initial conditions then $\alpha=\beta=1$.
So $a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
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} |
Equation with integers $x$, $y$ If $x$, $y$ positive integers ($x<y$), how can I solve the equation $x+y=14\sqrt{xy-48}$ ?
| Say $z=x+y\geq 0$ then $$ z^2 = 14^2(zx-x^2)-14^2\cdot 48$$ so $z= 14t$ and thus we have $$t^2= 14tx-x^2-48$$ or $$t^2-14tx +49x^2 =48(x^2-1)$$ so $$(t-7x)^2= 48(x^2-1)$$ and now we have $t-7x = 12s$ for some integer $s$. So we have $$ 3s^2= x^2-1$$ If $3|x-1$ then $$x-1 =3u^2\;\;\;{\rm and} \;\;\;x+1 = v^2$$ where $u,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Calculate the limit: $\lim_{x\to+\infty}(\frac{x^2 -x +1}{x^2})^{\frac{-3x^3}{2x^2-1}}$ without de l'Hôpital rule I was wondering how can I calculate the limit:
$$\lim_{x\to+\infty}\left(\frac{x^2 -x +1}{x^2}\right)^{\frac{-3x^3}{2x^2-1}}$$
without de l'Hôpital rule.
I tried to reconduct the limit at the well known one... |
I thought it might be instructive to present an approach that uses only the squeeze theorem and elementary inequalities obtained using pre-calculus analysis. To that end we proced.
In THIS ANSWER
, I showed using only the limit definition of the exponential function and Bernoulli's Inequality that the logarithm fun... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2574864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
$x^3 + 2x^2 + 5x + 2\cos x = 0$
$x^3 + 2x^2 + 5x + 2\cos x = 0$
How do I find the number of solutions of this equation (in $[0, 2\pi]$) without a graph?
Attempt:
The equation simplifies to $x(x^2 + 2x + 5)=- 2\cos x $
Minima of the quadratic occurs at $x= -1$ and it's value is $4$
Minima of $-2\cos x$ is $-2$
| $$y_1=-x^3-5x$$ and $$y_2=2x^2+2\cos x$$
for $x\ge0, y_1<0, y_2>0$
for $-1<x<0$ both function decrease and have only one intersection
for $x\le-1$ there are no intersection $$-x^3-5x<2x^2+2\cos x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Does $\sum_{k=0}^{2n}(-1)^k{2n\choose k}2^kF_{k+1}=5^n$ hold for all n values? I was looking at formula $(81)$ on here which is shown below
$$\sum_{k=0}^{n}{2n\choose k}2^kF_k=F_{3n}$$.
$F_n$ is the $n^{th}$ Fibonacci number
I wrote out the sum and just alternate the signs and found out that it has a simple answer in t... | Yes, the identity holds and is easily proven with induction on $n$. We first recall the familiar identity $$\binom{n}{k} + \binom{n}{k+1} = \binom{n+1}{k+1},$$ which is of course simply Pascal's triangle. If we apply this twice, we get $$\binom{n}{k-1} + 2\binom{n}{k} + \binom{n}{k+1} = \binom{n+2}{k+1},$$ which is e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many numbers can you get by multiplying two or more distinct members of the set $\{1,2,3,5,11\}$ together? Apparently the answer is 15, but I got 26.
My process:
The set $\{1,2,3,5,11\}$ has five numbers.
Ways to choose two members: $_5C_2$
Ways to choose three members: $_5C_3$
Ways to choose four members: $_5C_4$... | Note that, because two or more members can be multiplied, multiplying by $1$ will only make a difference if it is one of two numbers. Thus, multiplying by $1$ adds four potential numbers.
Now, we only need to consider the number of combinations that can be made from $2$, $3$, $5$, and $11$.
Choosing two from this set o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2579373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Factoring 2 in $\mathbb{Q}(\sqrt{-7})$ $7$ is a Heegner number, so all numbers in $\mathbb{Q}(\sqrt{-7})$ have a unique factorization.
I'm told that:
*
*$2$ is not prime in $\mathbb{Q}(\sqrt{-7})$,
*$3$ is not prime in $\mathbb{Q}(\sqrt{-11})$,
*$5$ is not prime in $\mathbb{Q}(\sqrt{-19})$,
*$11$ is not ... | Your mileage may vary...
I find these problems much easier if instead of trying to solve $(a - b \theta)(a + b \theta)$ (where $\theta = \frac{1 + \sqrt d}{2}) = p$, I try to solve $$\left( \frac{a - b \sqrt d}{2} \right) \left( \frac{a + b \sqrt d}{2} \right) = p.$$ Then $$\left( \frac{a - b \sqrt d}{2} \right) \left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Extremal problem with positive integer numbers Let $a,b$ be two positive integer numbers such that $a\sqrt{3}-b\sqrt{7}>0$. Find the minimum value of
$$
S=(a\sqrt{3}-b\sqrt{7})(a+b).
$$
Attempt I have tried and guess that the minimum value of $S$ is $(55+36)(55\sqrt{3}-36\sqrt{7})$, where $(55,36)$ is the integer solut... | You can always find solutions $(a,b)$ that give smaller values of $S$.
Consider solutions to the equation $x^2-21y^2=1$, $(x_1,y_1)=(55,12),(x_2,y_2)=(6049,1320)$ solutions from $(x_3,y_3)$ onward can be generated using solutions to the recursive solution $$x_{k+1}=x_1x_k+21y_1y_k\\y_{k+1}=x_1y_k+y_1x_k$$
Solutions th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Contest Inequality - Is it AM GM? To prove:
$$(a^2+2)(b^2+2)(c^2+2)\ge9(ab+bc+ca)$$
where $a,b,c$ are positive real numbers.
When does equality hold?
I thought equality would hold when $a = b = c$, but that doesn't seem to fit the statement.
To prove the inequality, I tried substituting $a = \tan A, b=\tan B$ and $c=\... | By C-S $$\left(\frac{(a+b)^2}{2}+1\right)(2+c^2)\geq(a+b+c)^2$$ and since
$$(a^2+2)(b^2+2)\geq3\left(\frac{(a+b)^2}{2}+1\right)$$ it's $$(a-b)^2+2(ab-1)^2\geq0,$$ we obtain:
$$(a^2+2)(b^2+2)(c^2+2)\geq3(a+b+c)^2\geq9(ab+ac+bc).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2582622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Different answers on $\frac{1}{2\pi}\int_0^{2\pi} \frac{1}{e^{ix}+c}\ dx$ where $c\in\mathbb{C}$ is fixed Upon the substitution $z=e^{ix}$, I get that this is equal to
$$\frac{1}{2\pi i} \oint_{C} \frac{dz}{z(z+c)},$$
where the integral is over the unit circle. I then split this up into
$$\frac{1}{2c\pi i}\left(\oint_{... | If $c=0$, then
$$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{dx}{e^{ix}+c}dx=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ix}dx = 0.
$$
If $c\ne 0$, then
\begin{align}
\frac{1}{2\pi}\int_{0}^{2\pi}\frac{dx}{e^{ix}+c}&=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{\frac{1}{ie^{ix}}d(e^{ix})}{e^{ix}+c} \\
&= \frac{1}{2\pi i}\int_{|z|=1} \frac{dz}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2585091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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limit of a trigonometric function $ \lim\limits_{x\to\pi/3} \frac{1 - 2\cos (x)}{\sin (3x)} $ compute the limit of
$$ \lim_{x\to \frac{\pi }{3} } \frac{1 - 2 \cos (x)}{\sin (3x)} $$
I would like to not do a translation with the change of variable $ t = x - \frac{\pi }{3} $
| $$\frac{1 - 2 \cos x}{\sin (3x)}=\frac{1- 2 \cos (x)}{-4\sin^3 x+3\sin x}=\frac{1}{\sin x}\frac{1- 2 \cos (x)}{3-4\sin^2 x}=\frac{1}{\sin x}\frac{1- 2 \cos (x)}{3-4+4\cos^2 x}=\frac{1}{\sin x}\frac{1- 2 \cos (x)}{4\cos^2 x-1}=\frac{1}{\sin x}\frac{1- 2 \cos (x)}{(2\cos x-1)(2\cos x+1)}=\frac{1}{\sin x}\frac{-1}{(2\cos ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2585787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 1
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To find the Value of $\tan A + \cot A$, if the value of $\sin A + \cos A$ is given To Find -
$$\tan A + \cot A$$
Given,
$$\sin A + \cos A = \sqrt2$$
My progress as far -
1st way-
$$\Rightarrow \sin A = \sqrt2 - \cos A$$
$$\Rightarrow \tan A = \frac{\sqrt2 - \cos A}{\cos A}$$
$$\Rightarrow \tan A = \frac{ \sqrt 2 }{\co... | $\cos\frac {\pi}4=\sin\frac {\pi}4=\frac1{\sqrt{2}}$. Hence what you have got is $$\cos\frac {\pi}4\sin A+\sin\frac {\pi}4\cos A=1$$ or $\sin\Big(A+\frac {\pi}4\Big)=1$. Can you solve from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In a square ABCD with side 14cm, 2 quadrants were made with centres A & B respectively and AB as radius. Find area of region I and II https://photos.app.goo.gl/5ibXDN5u6s0yo6KB3
I could do the following:
II + III = $\frac{1}{4}$ × $π$ × $14^2$ = $49π$ = $154cm^2$
II + IV = $\frac{1}{4}$ × $π$ × $14^2$ = $49π = 154 cm^2... |
Start by constructing $\overline {AE}$ and $\overline {EB}$. $E$ is the intersection of the arcs created by $ABD$ and $BAC$.
Notice that $\overline{AE} = \overline{EB} = \overline{AB} = 14$ because they are all radii of the circle. The points $A$, $B$ and $E$ form an equilateral triangle $\triangle ABE$. Hence, we kno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Find: $\lim_{x\to -\infty} f(x)= \frac{x^2-\sqrt{x^4+1}}{x^3-\sqrt{x^6+1}}$ (no L'Hopital)
Find: $\displaystyle \lim_{x\to -\infty} f(x)= \frac{x^2-\sqrt{x^4+1}}{x^3-\sqrt{x^6+1}}$ (no L'Hopital)
After developing the expression, by multiplying the fraction by the conjugates and rearranging, I found:
$$f(x)=x\times \f... | You certainly forgot an absolute value in your computations.
Here is a computation for $x<0$:
\begin{align}
f(x)&= \frac{x^2-\sqrt{x^4+1}}{x^3-\sqrt{x^6+1}}=-\frac 1{\Bigl(x^3-\sqrt{x^6+1}\Bigr)\Bigl(x^2+\sqrt{x^4+1}\Bigr)}\\
&= -\frac 1{\biggl(x^3-|x^3|\sqrt{1+\dfrac1{x^6}}\mkern2mu\biggr)\Bigl(x^2+\sqrt{x^4+1}\Bigr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 6
} |
Sum of all Fibonacci numbers $1+1+2+3+5+8+\cdots = -1$? I just found the sum of all Fibonacci numbers and I don't know if its right or not.
The Fibonacci sequence goes like this : $1,1,2,3,5,8,13,\dots$ and so on
So the Fibonacci series is this $1+1+2+3+5+8+13+\dots$
Let $1+1+2+3+5+8+\dots=x$
$$\begin{align}
1 + 1 + 2 ... | The problem is that the series you're trying to sum is divergent. You cannot manipulate divergent series by rules you can use with absolutely convergent series! Otherwise, by following the same "method" as yours, I can also claim the ridiculous statement that $1+2+3+\cdots=0$ as follows:
$$\begin{align}
1 + 2 + 3 + 4 +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Some infinite series (just for fun!) I have a few infinite series problems that I think MSE might enjoy, whose answers I already know:
$$\sum_{n=0}^\infty \frac{2^{n-2^n}}{1+2^{-2^n}}=\text{?}$$
$$\sum_{n=0}^\infty \frac{4^{n+2^n}}{(1+4^{2^n})^2}=\text{?}$$
$$\sum_{n=0}^\infty \frac{3^n(1+2\cdot3^{-3^n})}{2\cosh(3... | The tag telescoping is a great hint:
\begin{align*}
A_n &:= \frac{2^n}{2^{2^n}-1}
&\Rightarrow & \qquad A_n - A_{n+1} = \frac{2^n}{2^{2^n}+1} \\
B_n &:= \frac{4^n \cdot 4^{2^n}}{(4^{2^n}-1)^2}
&\Rightarrow & \qquad B_n - B_{n+1} = \frac{4^n \cdot 4^{2^n}}{(4^{2^n}+1)^2} \\
C_n &:= \frac{3^n}{3^{3^n}-1}
&\Rightarrow &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2592425",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Find all ordered pairs such that $x + y^2 = 2$ and $y + x^2 = 2$ How can I find all ordered pairs $(x,y)$ such that $x + y^2 = 2$ and $y + x^2 = 2$?
A solution/explanation would be greatly appreciated.
If it helps, the solutions given in the textbook are
x=-2, y=-2; x=1, y=1; x= (1+root 5)/2, y=(1-root 5)/2; ... | If we leave $y$ alone in the second equation, we have $y = 2-x^2$. Then putting it in the first equation, we get $x+(2-x^2) = 2 \implies x^4-4x^2+x+2 = 0$. Now, notice that $x=1$ satisfies this equation, therefore we can say that the polynomial $x^4-4x^2+x+2$ can be factorized as
$$x^4-4x^2+x+2 = (x-1)(Ax^3+Bx^2+Cx+D)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Integration using only two rules and by parts. I need to integrate a few fractions of form $\frac{p(x)}{q(x)}$ where ${p(x)}$ is a polynomial of degree 1 or and $q(x)$ being a polynomial of degree 2 only using integration by parts and following two rules:
$$\int \frac{f'(x)}{f(x)}\, dx = \ln| f(x)|+C$$
$$\int \frac{a}{... | $$I=\int\frac{3x-1}{x^2-2x+5}$$
$$\int\frac{ 2x-2}{(x-1)^2+4}+\int\frac{ x+1}{(x-1)^2+4}$$
$$\int\frac{( 2x-2)dx}{(x-1)^2+4}+\int\frac{ (z+2)dz}{z^2+4}$$
$$I=\frac 3 2\int\frac{ (2z)dz}{z^2+4}+\int\frac{ 2dz}{z^2+4}$$
Where $z=x-1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Find $\int^{1}_{0}x^{x}dx$ The answer can be in the form of two defined constants:
$$A = \frac{1}{1^1}+\frac{1}{2^2} + \frac{1}{3^3} + \cdots$$
$$B = \frac{1}{2^2} + \frac{1}{4^4} + \frac{1}{6^6} + \cdots $$
I would highly appreciate it if you included your thought process in the answer itself.
As pointed out by zz20s... | I have written a detailed explanation to find the integral of a general expression of the form $f(x) =x^{cx^a} $ from $0$ to $1$ here.
$$\int_{0}^{1} x^{cx^a}\, dx = 1- \frac{c}{(a+1)^2} + \frac{c^2}{(2a+1)^3} - \frac{c}{(3a+1)^4}+\ldots$$
On substituting $c =1$ and $a=1$, we get the desired result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Is this substitution to solve an integral correct? To solve the integral:
$\int\frac {x}{\sqrt{4-x^2}}\ dx$
I used the substitution:
$u = \sqrt{4-x^2}$
hence:
$\frac{du}{dx} = -\frac {x}{\sqrt{4-x^2}}$
$dx = -\frac {\sqrt{4-x^2}}{x}\ du$
So the integral becomes:
$\int\frac {x}{u}(-\frac {\sqrt{4-x^2}}{x})\ du$
$= \int\... | The two have more in common than they first seem to because $u=\sqrt{4-4\sin^2 t}=\cos t,\,du=-\frac{x}{2}dt$. A trigonometric substitution is so useful in such a variety of problems it's become a standard part of the toolkit.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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For triangle $ABC$ there are median lines $AH$ and $BG$ with $\angle CAH=\angle CBG={{30}^{0}}$ . Prove that $ABC$ is the equilateral triangle. For triangle $ABC$ there are median lines $AH$ and $BG$ with $\angle CAH=\angle CBG={{30}^{0}}$. Prove that $ABC$ is the equilateral triangle.
| Since $\measuredangle GBH=\measuredangle GAH$, we see that $ABHG$ is cyclic.
Thus, by Ptolemy $$AB\cdot GH+BH\cdot AG=AH\cdot BG$$ or in the standard notation
$$\frac{c^2}{2}+\frac{ab}{4}=\frac{1}{4}\sqrt{(2b^2+2c^2-a^2)(2a^2+2c^2-b^2)},$$
which after squaring of the both sides gives
$$(a-b)^2(a+b-c)(a+b+c)=0$$ or
$$a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Solve $(5+2\sqrt{6})^{\frac{x}{2}} + ( 5-2\sqrt{6})^{\frac{x}{2}} = 10$ I wish to solve the equation
$$(5+2\sqrt{6})^{\frac{x}{2}} + ( 5-2\sqrt{6})^{\frac{x}{2}} = 10$$
I tried factorizing until I reached
$(\sqrt{2}+\sqrt{3})^x + (\sqrt{2}-\sqrt{3})^x = 10$
But from there I don't know what to do any help would be welc... | There are two solutions: $x=2$ and $x=-2$.
We easily see that $x=2$ is a solution. There are no other solutions $x>0$ because the left-hand side is an increasing function on ${\mathbb R}^+$.
Indeed, noticing that $1/(5+2\sqrt{6})=5-2\sqrt{6}$, we then find that
$$
f(x) = (5+2√6)^{\frac{x}{2}} + ( 5-2√6)^{\frac{x}{2}}
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Alternating summation and subtraction of square roots I encountered a problem, to find the integer part of: $$\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{3} + \sqrt{4}} +...+\frac{1}{\sqrt{99} + \sqrt{100}}$$.
I multiplied the conjugate of each denominator. Meaning, for $\frac{1}{\sqrt{a} + \sqrt{b}}$, I multiply $... | We want to find the integer part of
$\sum_{x=1}^{50} f(x)$
where
$f(x) = \sqrt{2x}-\sqrt{2x-1}$.
For decreasing $f(x)$,
$$\int_a^{b+1} f(x) \; dx < \sum_{x=a}^b f(x) < \int_{a-1}^b f(x) \; dx$$
(It may help to make a sketch to see this.)
It's easier to work with the sum from $x=2$ to $50$ instead of $x=1$ to 50 becau... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2600710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Arithmetic and harmonic mean of two numbers. What is the maximum value of arithmetic mean of two integer number if their harmonic mean is 40?
$$\frac{1}{x}+\frac{1}{y} = \frac{2}{40}$$
$$xy=20x+20y$$
$$xy−20x−20y+20⋅20=400$$
$$(x−20)(y−20)=400$$
| $$\frac{1}{x}+\frac{1}{y} = \frac{2}{40}$$
$$xy=20x+20y$$
$$xy−20x−20y+20⋅20=400$$
$$(x−20)(y−20)=400$$
$400=1\cdot 400 =2\cdot 200 =4\cdot 100=5\cdot 80 = 8\cdot 50= 10\cdot 40=16\cdot 25=20\cdot 20$
$$ x+y = \{21+420; \;22+220;\;24+120;\;25+100;\;28+70;\;30+60;\;36+45;\;40+40\}
$$
the maximum value of summ is 441 an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2604238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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System of simultaneous equations involving integral part (floor) Solve the following system of simultaneous equations:
$$
\left \{
\begin{matrix}
x^2+[y]=10 \\
y^2+[x]=13
\end{matrix}
\right .
$$
where square brackets $[...]$ denote integral part of a number (a.k.a. floor)
This is what I've come to:
Case I. $y\ge 0$.
... | Hint: $\;\lfloor x \rfloor \le x \lt \lfloor x \rfloor + 1\,$ and the same for $\,y\,$, then:
$$
\left \{
\begin{matrix}
10 \le x^2+y \lt 11 \\
13 \le y^2+x \lt 14
\end{matrix}
\right .
\;\;\implies\;\; 23 \;\le\; x^2+y+y^2+x \,=\, (x+1/2)^2+(y+1/2)^2 - 1/2 \;\lt\; 25
$$
It follows that $|x+1/2|, |y+1/2| \lt \sqrt{25+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2604419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determine the infinitesimal order of the function in $x0= 0$ Problem: infinitesimal order. If you can explain me how you solve this
function:
$$(1+3x)^x -1 -x\ln(1+3x)$$
Mine resolution: $$(1+3x)^x$$ = 1 (taylor series)
$$xln(2+3x)$$ = $3x^2$
so $$lim \frac {1-1-3x^2}{x^n}$$= -3 (this is not equal to... | Note that
$$x\ln(1+3x)=x\left(3x-\frac{9}{2}x^2+9x^3+o(x^3)\right)=3x^2-\frac{9}{2}x^3+9x^4+o(x^4)$$
$$(1+3x)^x=e^{x\ln(1+3x)}=e^{3x^2-\frac{9}{2}x^3+9x^4+o(x^3)}=1+3x^2-\frac{9}{2}x^3+9x^4+\frac{(3x^2-\frac{9}{2}x^3+9x^4+o(x^4))^2}{2}+o(x^4)=1+3x^2-\frac{9}{2}x^3+9x^4+\frac{9}{2}x^4+o(x^4)$$
thus
$$(1+3x)^x -1 -x\ln(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2606439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $a^2+b^2+c^2\gt \frac {1}{2018}$ given $\left({3a + 28b + 35c}\right)\left({20a + 23b +33c}\right) = 1$ Let $a, b, c$ be real numbers such that $\left({3a + 28b + 35c}\right)\left({20a + 23b +33c}\right) = 1$. Prove that $a^2+b^2+c^2\gt \frac {1}{2018}$.
It looks like an easy question, but I thought for a while a... | With a bit of linear algebra:
An equivalent problem is
$$\max_{(a,b,c)\ne 0} \frac{(3a+28b+35c)(20a+23b+33c)}{a^2 + b^2 + c^2}$$
or, equivalently
$$\max_{a^2 + b^2 + c^2 =1} (3a+28b+35c)(20a+23b+33c)$$
The symmetric quadratic form $$(a,b,c) \mapsto (3a+28b+35c)(20a+23b+33c)=\\=60 a^2 + 629 a b + 799 a c + 644 b^2 + 17... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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$x+\frac{1}{x}$ is an integer Is the following Proof Correct? In particular please comment on the correctness of the given formulas.
Theorem. Given that $x$ is a real number, $x\neq 0$, and $x + \frac{1}{x}$ is an integer. For all $n\ge 1$, $x^n+\frac{1}{x^n}$ is an integer.
Proof. We construct the proof by recourse to... | I think it is easier to see that
$$
x^{n+2}+\frac{1}{x^{n+2}}=\left(x+\frac{1}{x}\right)\left(x^{n+1}+\frac{1}{x^{n+1}}\right)-\left(x^n+\frac{1}{x^n}\right)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Volume of Region Bounded by $y = \sqrt{x}$ and $y = 18-x^2$ I haven't a clue why I'm having so much issues solving this problem. The two curves are $y=\sqrt{x}$ and $y=18-x^2, x \ge 1$, and rotated about the $y$-axis.
If I'm not mistaken, I need to use the shell method. In that case, the volume integration formula is $... | Note that the graph of $y = \sqrt{x}$ and $y = 18-x^2$ shows that these two curves meet at $x = 4$ and $y = 18-4^2 = \sqrt{4} = 2$. Write out explicitly $f$ in the question body.
$$f(x) = 18-x^2-\sqrt{x} \quad \forall\,x \in [1,4]$$
\begin{align}
\text{Required volume of revolution}
&= 2\pi \int_1^4 x f(x) dx \\
&= 2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Maclaurin series questions How can I find the Maclaurin series for $\ln(1+x^2+x)$? I'm not sure if I should try writing 1+x+x^2 in some other way or..Also if I want to find $h^{(3k)}(0)$ (the derivative of order 3k) where should I start?
| Take $f(x)=\ln(x^2+x+1)$ therefore $f'(x)=\frac{2x+1}{1+x+x^2}=\frac{1+x-2x^2}{1-x^3}$. Also we have:$$\frac{1}{1-x^3}=\sum_{n=0}^{\infty}x^{3n}$$then we have$$f'(x)=(1+x-2x^2)\sum_{n=0}^{\infty}x^{3n}=\sum_{n=0}^{\infty}x^{3n}+\sum_{n=0}^{\infty}x^{3n+1}+\sum_{n=0}^{\infty}-2x^{3n+2}=\sum_{n=0}^{\infty}a_nx^n$$where $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Can we prove the same cardinality of the sets $\mathbb{N}$ and $\mathbb{N^2}$ this way? I tried to prove that the sets $\mathbb{N}$ and $\mathbb{N^2}$ have the same cardinality and I concluded the following:
Consider the function $f:\mathbb{N}\rightarrow \mathbb{N^2}$ that achieves the mapping:
$ \begin{cases} 1 \righ... | With $0 \in \mathbb N$ we trace out a path:
$(0,0) \to$
$(1,0) \to (1,1) \to (0,1) \to $
$(2,0) \to (2,1) \to (2,2) \to (1,2) \to (0,2) \to $
$(3,0) \to (3,1) \to (3,2) \to (3,3) \to (2,3) \to (1,3) \to (0,3) \to $
$\text{etc.} \qquad \text{Figure 1}$
Here is the corresponding mapping:
$$
\pi(m,n) = \left\{\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Proof verification: If $a \equiv b (mod$ $n)$, then $a^3 \equiv b^3 (mod$ $n)$ Would someone be willing to verify the following proof?
Theorem: Suppose $a, b \in \mathbb{Z}; n \in \mathbb{N}$. If $a \equiv b (mod$ $n)$, then $a^3 \equiv b^3 (mod$ $n)$.
Proof:
$a \equiv b (mod$ $n) \rightarrow xn = a - b; x \in \mathb... | The posted proof is correct. It might be even easier, however, to first prove the more general:
$$
a \equiv b \pmod{n} \quad\text{and}\quad a' \equiv b' \pmod{n} \quad \implies \quad a \cdot a' \equiv b \cdot b' \pmod{n}\quad
$$
The proof goes the same: $a\cdot a' = (xn+b)\cdot(x'n+b')=b\cdot b'+n\cdot (x\cdot b'+ x' \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.