Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Condition for the generators through any one of the ends of an equi-conjugate diameter of the principal elliptic section of the hyperboloid
Question: Show that the generators through any one of the ends of an equi-conjugate diameter of the principal elliptic section of the hyperboloid $\frac{x^2}{a^2}+\frac{y^2}{b^2}... | $\alpha = 60^\circ$ is the angle between two generators not the parameter of end points of conjugal diameters.
$$\begin{align}
\cos\alpha &=\frac{l_1l_2+m_1m_2+n_1n_2}{\sqrt{{l_1}^2+{m_1}^2+{n_1}^2}\sqrt{{l_2}^2+{m_2}^2+{n_3}^2}} \\
\implies \cos\alpha &=\frac{a^2\sin^2\theta+b^2\cos^2\theta-c^2}{a^2\sin^2\theta+b^2\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2533581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $a,b,c$ are three complex numbers Find possible values of $\lvert a+b+c \rvert$ Given three complex numbers $a,b,c$ such that $\lvert a \rvert=\lvert b \rvert=\lvert c \rvert=1$ and $$\frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}=-1$$
Find which of the following are possible values of $\lvert a+b+c \rvert$
A)0
B)2
C)... | We will use some facts such as $|a|=1$ implies $a\bar{a}=1$. So $\frac{1}{a}=\bar{a}$. Also let $w=a+b+c$, then $|w|^2=(a+b+c)(\bar{a}+\bar{b}+\bar{c})$.
\begin{align*}
\frac{a^2}{bc}+\frac{b^2}{ac}+\frac{c^2}{ab}&=-1\\
\frac{a^3}{abc}+\frac{b^3}{abc}+\frac{c^3}{abc}&=-1\\
a^3+b^3+c^3+abc & =0\\
a^3+b^3+c^3-3abc & =-4a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2533698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Only certain values seem to make an expression into perfect square, can you help me prove or disprove it? All the variables in this question are in integers. I am trying to prove that
$$\frac{4k-\Delta^2}3$$
is a perfect square only if $\Delta \in \{\pm(2a+b),\pm(a+2b),\pm(a-b)\}$ where $a,b$ are such that $k=a^2+ab+b^... | Suppose $\frac{4k - \Delta^2}{3}$ is a square, in particular say $$\frac{4k - \Delta^2}{3}=M^2.$$ Then $4k = \Delta^2 + 3 M^2$.
Reducing modulo 4, we see that $\Delta^2 + 3M^2 = 0 \pmod{4}$, so $\Delta^2 = M^2 \pmod{4}$. Hence, $\Delta=M \pmod{2}$, so they are both even or both odd.
If they are both odd:
Set $$a = \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2537381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Closed-form expression $(x_{2^l})_{l\in \mathbb{N}_0}$ I have this problem I'm not sure how to solve.
Given $x \in \mathbb{R}^{\mathbb{N}_0}_{\geq 0}$ and
$x_k=\left\{\begin{matrix} 0 & k=0\\ 4x_{\left \lfloor \frac{k}{2}
> \right \rfloor} + k^2 & k \in \mathbb{N} \end{matrix}\right.$
Determine a closed-form expressi... | Your calculation should be revised. We obtain for $l\in\mathbb{N}_0$:
\begin{align*}
x_{2^{l+1}}=4x_{2^l}+\color{blue}{\left(2^{l+1}\right)^2}=4x_{2^l}+4^{l+1}
\end{align*}
We start with $k=2^l$, calculate a few iterations and look if we can make a good guess of a closed formula. The validity of the formula co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2538079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find Jordan basis for the following nilpotent matrix I need to find the Jordan basis and Jordan form for this nilpotent matrix:
$$\begin{pmatrix}
-2i & 1 & 10+5i \\
4 & 2i & -10+19i \\
0 & 0 & 0 \\
\end{pmatrix}$$
The matrix is nilpotent so all eigenvalues are 0, therefore $Av_1=0$ and from that I got that $v_1=\begin{... | You have it a bit backwards. We have $A^3 = 0$ but $A^2 \neq 0.$ It appears you want the columns of the change of basis matrix, call that $V,$ as you want column vectors $v_1, v_2,v_3.$
Well,
$$
A^2 =
\left(
\begin{array}{rrr}
0 & 0 & -i \\
0 & 0 & 2 \\
0 & 0 & 0
\end{array}
\right)
$$
In order to arrange $A^2 v_3 \n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2538576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the greatest n for which the sum $\sum_{k=1}^n \lfloor\sqrt{k}\rfloor$ is a prime number. I've come across the sum:
$$\sum_{k=0}^n \lfloor\sqrt{k}\rfloor$$
According to Find a formula for $\sum\limits_{k=1}^n \lfloor \sqrt{k} \rfloor$
I approached the formula:
$$\sum_{k=0}^n \lfloor\sqrt{k}\rfloor = n\lfloor\sqrt{... | Note that for $n\geq 1$, the sum $S_n$ is a positive integer which can be written as
$$S_n:=\sum_{k=0}^n \lfloor\sqrt{k}\rfloor =\frac{1}{6}\cdot\lfloor\sqrt{n}\rfloor\cdot\left(6n-(2\lfloor\sqrt{n}\rfloor^2+3\lfloor\sqrt{n}\rfloor-5)\right).$$
where $f_1:=\lfloor\sqrt{n}\rfloor$ and $f_2:=\left(6n-(2\lfloor\sqrt{n}\rf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539309",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding value of $c+d$
At a point $A(1,1)$ on the ellipse , equation of tangent is $y=x.$ If one of the
foci of the ellipse is $(0,-2)$ and coordinate of center of ellipse is $(c,d)$.
Then find value of $c+d$ (given length of major axis is $4\sqrt{10} unit$)
Attempt : assuming one foci is at $S_{1}(0,-2)$ and other i... | We have
*
*Sum of distances from focii $=2a = 4\sqrt{10}$ from which we obtain
$$(\alpha-1)^2+(\beta-1)^2 = 90$$
*Product of distances from the tangent (which is helpfully $x=y$) is $b^2$ ($b$ is the semi-minor axis) and hence $$4(\alpha-\beta)=4b^2$$ (Noting that $\alpha>\beta$)
*Distance between foci $=\sqrt{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Finding roots of a specific Function How can I calculate the roots of this equation? What about in MATLAB?
x2=N(N(x2))
The plot of the function is in below:
| If I understand correctly, $N$ is the function defined piecewise by
$$
N(x) = \begin{cases}
5 & x \leq 2, \\
9 - 2x & 2 < x < 4, \\
1 & x \geq 4.
\end{cases}
$$
We can then calculate what $N(N(x))$ is.
*
*If $x \leq 2$ then $N(x) = 5$, so $N(N(x)) = N(5) = 1$.
*If $x \geq 4$ then $N(x) = 1$, so $N(N(x)) = N(1) = 5$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that any field $F$ containing $\sqrt{a}+\sqrt{b}$ also contains $\sqrt{a}$ and $\sqrt{b}$.
Prove that any field $F$ containing $\sqrt{a}+\sqrt{b}$ also contains $\sqrt{a}$ and $\sqrt{b}$.
I started by taking $(\sqrt{a}+\sqrt{b})^2=2\sqrt{ab}+a+b$ and want to conclude that $\sqrt{ab}\in F$ but am unsure of this.... | You have a good start by showing that $a \sqrt{b} + b\sqrt{a} \in F$, but this does not show that $x\sqrt{a} + y\sqrt{b} \in F$ for an arbitrary choice of $x$ and $y$ ($a$ and $b$ are fixed constants).
What you want to do is add something to $a \sqrt{b} + b\sqrt{a}$ to cancel out one of the roots. Like maybe a multip... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Find the range of eccentricity of an ellipse such that the distance between its foci doesn't subtend any right angle on its circumference.
What is the range of eccentricity of ellipse such that its foci don't subtend any right angle on its circumference?
I thought that the eccentricity would definitely be more than $... | Your answer is correct, except that $0$ should be included (there are no right angles subtended in a circle).
Here's a complete solution:
Using the parameterization $P=(a \cos\theta, b\sin\theta)$ for an origin-centered ellipse with major radius $a$ (in the $x$ direction) and minor radius $b$ (in the $y$ direction), c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proving that $a_n=\sqrt{n^2+n}-n$ converges, finding its limit and showing that its sequence ${(a_n)^{\infty}_{n=1}}$ is monotinic. As the title says, below I've proved the statement. Just posted here for verification and correction! And of course for other people to be inspired.
Let $a_n=\sqrt{n^2+n}-n$ with $n\in\mat... | $$a_n={n\over \sqrt{n^2+n}+n}= {1\over \sqrt{1+{1\over n}}+1}$$
so $$ 0<a_n\leq {1\over 2}$$
and since $\sqrt{1+{1\over n}}+1$ is decreasing so is $a_n$ increasing. So $a_n$ is convergent with limit ${1\over 2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Use Taylor’s Theorem to approximate $\sqrt{e}$ to within an error of magnitude at most $10^{-6}$ Could you please let me know if the following considered as a correct answer to this question.
Taylor Series
$f(x) = \sum_{k=0}^{n} \frac{f^{(k)}x_{0}}{k!}(x - x_{0})^{k}+ \frac{f^{(n+1)}(c)}{(n+1)!}(x - x_{0})^{(n+1)}$.
Ta... | $$E_7 \leq \frac{\sqrt{3}}{2^8 \cdot 8!} \approx 1.67803161 \times 10^{-7} < 10^{-6}$$
$n=7$ suffices.
Note that Showing $n =6$ is not sufficient doesn't imply that $n=7 $ is sufficient, we still have to verify that it works.
Code to check that $n=7$ is indeed the smallest number of terms required.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $a^{4} + b^{4} + a^{2} b^{2} = 60 \ \ a,b \in \mathbb{R}$ Then prove that : Let $a^{4} + b^{4} + a^{2} b^{2} = 60 \ \ a,b \in \mathbb{R}$ Then prove that :
$$4a^{2} + 4b^{2} - ab \geq 30$$
My attempt: :
$$4a^{2} + 4b^{2} - ab \geq 30 \\ 4(a^2+b^2)-ab \geq30 \\4(60-a^2b^2)-ab\geq30\\ 240-30\geq4(ab)^2+ab\\ 4(ab)^2+... | We have $$[4(a^2+b^2)]^2=16(a^4+b^4+2a^2b^2) =16(60+a^2b^2)$$
Then, $$ [4(a^2+b^2)]^2 - (30 +ab)^2 =16(60+a^2b^2) - (30 +ab)^2 \\=15a^2b^2 -60 ab - 60 =\color{red}{15(ab-2)^2\ge0} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2552804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Finding: $\lim\limits_{x\to -\infty} \frac{6x^2+5\cos{\pi x}}{\sqrt{x^4+\sin{5\pi x}}}$
I'm running into problems with this limit:
$$\lim_{x\to -\infty} \frac{6x^2+5\cos{\pi x}}{\sqrt{x^4+\sin{5\pi x}}}$$
I've tried using l'Hospitals rule, however we will alway keep the $\cos(\pi x)$ expression, as well for $\sin(5... | Writing out like
$$\frac{6x^2+5\cos\pi x}{\sqrt{x^4+5\sin 5\pi x}} = \frac{x^2\left(6 + \frac{5\cos \pi x}{x^2}\right)}{x^2\sqrt{1 + \frac{5\sin 5\pi x}{x^4}}}$$
should do the trick.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2555213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Calculate the derivative using limit definition. This is the function $f(x)$$=\frac{1}{\sqrt{3x-2}}$ .
I wrote that $$\lim_{h\to 0}\frac{\frac{\sqrt{3x+3h-2}}{3x+3h-2}-\frac{\sqrt{3x-2}}{3x-2}}{h}.$$
I am not able to continue further.
| Let $a = \sqrt{3x+h-2}$ and $b = \sqrt{3x-2}$, then $a^2 - b^2 = h$ which means $h \to 0$ as $a \to b$. So we've:
$$ \lim_{a \to b} \frac{1/a-1/b}{a^2-b^2} = \lim_{a \to b} \frac{b-a}{(ab)(a+b)(a-b)} = - \lim_{a \to b} \frac{1}{ab(a+b)} = -\frac{1}{2 b^3} = -\frac{1}{2\sqrt{(3x-2)^3}.} $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2556339",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
If $x,y,z\in {\mathbb R}$, Solve this system equation: If $x,y,z\in {\mathbb R}$, Solve this system equation:
$$
\left\lbrace\begin{array}{ccccccl}
x^4 & + & y^2 & + & 4 & = & 5yz
\\[1mm]
y^{4} & + & z^{2} & + & 4 & = &5zx
\\[1mm]
z^{4} & + & x^{2} & + & 4 & = & 5xy
\end{array}\right.
$$
This is an olympi... | Summing up all the equtions you find:
$$\sum_{cyc} x_i^4 + \sum_{cyc} x_i^2 - 5\sum_{cyc} x_ix_j +12= 0$$
Now manipulate in such way to have the equality as sum of squares!
EG
$$-5xy =\frac52 \left( x-y\right)^2-\frac52x^2-\frac52y^2$$
That is:
$$x^4+y^4+z^4-4x^2-4y^2-4z^2+\frac52 \left( x-y\right)^2+\frac52 \left(y-z\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2556960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Solution of the ordinary differential equation $y'(t)=-y^3+y^2+2y$ Consider the solution of the ordinary differential equation $y'(t)=-y^3+y^2+2y$ subject to $y(0)=y_0 \in (0,2). $ then $\lim \limits_{t \to \infty}y(t)$ belongs to
*
*{-1,0}
*{-1,2}
*{0,2}
*{0, $+\infty $}
My Attempt: $y'(t)=-y^3+y^2+2y \\ \Rig... | Given that $$y'(t)=-y^3+y^2+2y=-y\left(y^2-y-2\right)=-y(y-2)(y+1)$$
Clearly at $~y=-1,~0,~2~$ we have $~y'=0~.$
So the direction field will be something like:
When $~y<-1~,$ slope of $~y~$ is increasing.
When $~-1<y<0~,$ slope of $~y~$ is decreasing.
When $~0<y<2~,$ slope of $~y~$ is increasing.
When $~y>2~,$ slope o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2563252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove the inequality $\frac{b+c}{a(y+z)}+\frac{c+a}{b(z+x)}+\frac{a+b}{c(x+y)}\geq \frac{3(a+b+c)}{ax+by+cz}$
Suppose that $a,b,c,x,y,z$ are all positive real numbers. Show that
$$\frac{b+c}{a(y+z)}+\frac{c+a}{b(z+x)}+\frac{a+b}{c(x+y)}\geq \frac{3(a+b+c)}{ax+by+cz}$$
Below are what I've done, which may be misleadi... | Partial Proof:
If we start with your substitution at 3) and remark that we have with your condition :
$$ux+yv+zw=(x+y+z)(u+v+w)-u(y+z)-v(z+x)-w(x+y)=(x+y+z)-u(y+z)-v(z+x)-w(x+y)$$
So if we put :
$p=y+x$$\quad$$i=w(x+y)$
$q=z+x$$\quad$$k=v(z+x)$
$r=y+z$$\quad$$j=u(y+z)$
We get :
$$\frac{1}{i}+\frac{1}{j}+\frac{1}{k}+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2564669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 3,
"answer_id": 1
} |
Quadratic equation with different indices My maths teacher gave me a worksheet to work through as I was getting slightly bored in lessons. However, there was one question which I cannot do. The worksheet gives the answer, but you are supposed to show how you did it. Here is the question:
$729 + 3^{2x+1} = 4\times3^{x... | Hint:
This seems to be a problem in arithmetic. Note that $729=3^6$, and rewrite the equation as
$$3^{2x+1}+3^6=4\cdot 3^{x+3}.$$
Some details
If $x$ is a natural number this implies $4$ divides the left-hand side. Depending on the values of $x$, factor out $3^6$ or $3^{2x+1}$. $4$ must divide the other factor. Now
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2565365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Sequence : $a_{n+1}=2a_n-a_{n-1}+2$ Let $c$ be a positive integer. The sequence $a_1, a_2, \ldots$ is defined by $a_1=1, a_2=c$ and
$a_{n+1}=2a_n-a_{n-1}+2$ for all $n \geq 2$.
Prove that for each $n \in \mathbb{N}$ there exists $k \in \mathbb{N}$ such that $a_na_{n+1} = a_k$.
My attempt :
Trying with small numbers,... | For $n=1$, $a_1a_2=c=a_2$.
In the following, $n\ge 2$.
You already have $a_n=(n-1)c+(n-2)^2$.
Then,
$$\begin{align}&a_na_{n+1}=a_k\\\\&\iff ((n-1)c+(n-2)^2)(nc+(n-1)^2)=(k-1)c+(k-2)^2\\\\&\iff k^2+(c-4)k-(c^2n^2-c^2n+2cn^3-7cn^2+7cn+n^4-6n^3+13n^2-12n)=0\\\\&\iff k=\frac{-c+4\pm\sqrt{\Delta}}{2}\end{align}$$
where
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2568850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Transformation matrix - Are the matrices correct? I want to describe for the following $\mathbb{R}$-vector spaces $V$with basis $B$ the matrix $A_{f, B, B}$ for $f\in \text{End}_K(V)$:
*
*$V=\mathbb{R}^2, B=((1,0)^T, (0,1)^T)$ and $f$ the rotation by 45 degrees clockwise.
*$V=\mathbb{C}, B=(1,i)$ and $f(z)=a^2\ov... | For the first, note that rotation matrix by $\theta$ clockwise is:
$$f(x)=\begin{pmatrix}\cos\ \theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$$
Thus:
$$f(x)=\begin{pmatrix}\cos\frac{\pi}{4} & \sin\frac{\pi}{4} \\ -\sin\frac{\pi}{4} & \cos\frac{\pi}{4}\end{pmatrix}\cdot x=\frac{1}{\sqrt{2}}\begin{pmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding the maximum of $p^3 + q^3 +r^3 + 4pqr$
$p$,$q$,$r$ are $3$ non-negative real numbers less than or equal to $1.5$ such that $p+q+r = 3$, what will be the maximum of $p^3 + q^3 + r^3 + 4pqr$ ?
I tried AM-GM on $p,q,r$ to get the maximum of $pqr$ as $1$, but on doing it for $p^3 + q^3 + r^3 $, I get the minimum... | Let $p+q+r=3u$, $pq+pr+qr=3v^2$ and $pqr=w^3$.
Hence, the condition does not depend on $w^3$ and we need to find a maximal value of $f,$ where
$$f(w^3)=27u^3-27uv^2+7w^3.$$
We see that $f$ increases,
which says that it's enough to solve our problem for the maximal value of $w^3$.
Now, $p$, $q$ and $r$ are three non-ne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Simplify $\frac{7^{ \log_{5} 15 }+3^{2+\log_{5}7}}{7^{\log_{5}3}}$ I know that the result of this expression is 16 but how do I get to that result?
$$\frac{7^{ \log_{5} 15 }+3^{2+\log_{5}7}}{7^{\log_{5}3}}$$
| $$\frac{7^{ \log_{5} 15 }+3^{2+\log_{5}7}}{7^{\log_{5}3}}=\frac{7^{ 1+\log_53 }+7^{\left(2+\log_{5}7\right)\log_73}}{7^{\log_{5}3}}=7+\frac{7^{\log_79+\log_53}}{7^{\log_53}}=7+9=16.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
How to prove $\{n!e\} = \frac{1}{n +1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \dots$ I have the definition $a_n = \frac{1}{n +1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} + \dots$
I need to show that:
$a)$ $0 < a_n < \frac{1}{n}$
$b)$ $a_n = n!e - \lfloor{n!e}\rfloor$
So I know that
$\frac{1}{n} ... | Because $$n!e=n!\left(2+\frac{1}{2!}+...+\frac{1}{n!}\right)+\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+...$$ and
$$\frac{1}{n+1}+\frac{1}{(n+1)(n+2)}+...<\frac{1}{n+1}+\frac{1}{(n+1)^2}+...=\frac{\frac{1}{n+1}}{1-\frac{1}{n+1}}=\frac{1}{n}<1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How I do evaluate this polynomial problem? Let $f(x)=x^2+ax+b$.
Suppose $f(f(x))=0$ equation has $4$ different real solutions $x_1,x_2,x_3,x_4$
and that two of them sum up to $-1$ (i.e. $\exists\, i\neq j$ such that $x_i+x_j=-1$)
Prove that $b\lt-\frac14$
| Let $y = f(x)$ then we have the following system
$$ \begin{cases} x^2 + ax + b = y & (1) \\ y^2 + ay + b = 0 & (2) \end{cases} $$
In order to have 4 distinct real solutions in $x$, we require two distinct real solutions in $y$, therefore $a^2-4b > 0$. Denote the two solutions as $y_1$ and $y_2$, then
$$ \begin{cases} x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2574097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
$(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=1$ is true if and only if $x+y=0$ Prove that $(x+\sqrt{x^2+1})(y+\sqrt{y^2+1})=1$ is true if and only if $x+y=0$
I believe x and y could both be 0 as that satisfies the equations. Beyond that, I do not know how to prove this.
| You have $$(x+\sqrt{x^2+1})(y+\sqrt{y^2+1}) = 1$$
$$x+\sqrt{x^2+1} = \sqrt{y^2+1} - y$$
$$x+y = -\sqrt{x^2+1} + \sqrt{y^2+1}$$
$$(x+y)(\sqrt{x^2+1} + \sqrt{y^2+1}) = -x^2+y^2$$
$$(x+y)(\sqrt{x^2+1} + \sqrt{y^2+1} +x-y) = 0$$
Now, you get $x+y=0$ or $\sqrt{x^2+1} + \sqrt{y^2+1} +x-y = 0$. Can you go further?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2574195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
More of this type $T_n^2+T_{n+1}^2+T_{n+2}^2=X^2?$ Given triangular numbers,
$$T_n:= {n(n+1)\over 2} = 1,3,6,10,15,...$$
Let $T_n, T_{n+1}$ and $T_{n+2}$ be three consecutive triangular numbers.
My question is: Are there more of $T_n^2+T_{n+1}^2+T_{n+2}^2$ that only yield a square number? Or this is $$T_3^2+T_4^2+T_5^... | I've checked this for all $X$ up to $10^{10^5}$, here's how.
We are searching for integer solutions to $T_{n-1}^2 + T_n^2 + T_{n+1}^2 = X^2$ (note I've reparametrised, compared to the OP), which expands to
$$3 n^4 + 6 n^3 + 15 n^2 + 12 n + 4 =4X^2. $$
Curiously enough, this is an invertible expression, and we can inst... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is this a valid method of solving $3^x + 2^x = 35$? I want to know if this is a valid method of solving this equation:
$3^x + 2^x = 35$
$3^x+2^x = (7)(5)$
$3^x+2^x = (3+2^2)(2+3)$
$3^x+2^x = 3^2 + 2^3 + 18$
$3^x+2^x = 9 + 18 + 2^3 $
$3^x+2^x = 27 + 2^3 $
$3^x+2^x = 3^3 + 2^3 $
And now comes my problem. Is it correct to... | $3$ it's an unique root because $f(x)=3^x+2^x$ is an increasing function and $f(3)=35$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
using a base 3 decimal to express as a base 10 fraction using geometric series
Express $0.\overline{21}_3$ as a base 10 fraction in reduced form.
So I was able to solve it by setting $x=\overline{.21}$, but the solution also briefly mentioned another way using the geometric series:
A quick way to get the answer by u... | You have
$(0.212121 \ldots)_3 = \frac{7}{9} + \frac{7}{81} + \frac{7}{729} + \dots$
and for $0<a<1$
$1+a+a^2+\ldots = \dfrac{1}{1-a}$
from which
$\begin {align}
(0.212121 \ldots)_3 &= 7\left(\frac{1}{9} + \frac{1}{81} + \frac{1}{729} + \dots
\right)\\
&= 7\left(\frac{1}{1-\frac{1}{9}} -1 \right)\\
&= 7\left(\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Evaluate $\int \frac{a^2\sin^2 x+b^2\cos^2 x}{a^4\sin^2 x+b^4\cos^2 x}dx$ Evaluate
$$\int \frac{a^2\sin^2 x+b^2\cos^2 x}{a^4\sin^2 x+b^4\cos^2 x}\text dx$$
I would have given my attempt to this question but honestly, I think my attempts to solve this did nothing but only complicated it further.
Any hints or suggestions... | Hint:
We have $$I = \int \frac{a^2\sin^2 x + b^2\cos^2 x}{a^4\sin^2 x + b^4\cos^2 x}\, dx = \int \frac{[a^2-b^2]\sin^2 x + b^2}{[a^4-b^4]\sin^2 x + b^4}\, dx $$ $$= (b^2-\frac{(a^2-b^2)b^4}{a^4-b^4})\int \frac{1}{(a^4-b^4)\sin^2 x + b^4}\, dx + \int \frac{a^2-b^2}{a^4-b^4} \, dx$$
The first integral can be easily calcu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Why is my alternate method of calculating scalar products not working? The exercise is such: Given that $|\vec{a}| = 3$, $|\vec{b}| = 2$ and $\varphi = 60^{\circ}$ (the angle between vectors $\vec{a}$ and $\vec{b}$), calcluate scalar product $(\vec{a}+2\vec{b}) \cdot (2\vec{a} - \vec{b})$.
My initial thought was to sol... | I think it is failed because here
$(\vec{a} + 2\vec{b}) \cdot (2\vec{a} - \vec{b}) = |\vec{a} + 2\vec{b}| \cdot |2\vec{a} - \vec{b}| \cdot \cos60^{\circ} = 5 \cdot 2\sqrt{7} \cdot \frac{1}{2} = 5\sqrt{7}$
you are assuming that the angle between vectors $(\vec{a} + 2\vec{b})$ and $(2\vec{a} - \vec{b})$ is $60^\circ$ but... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2578792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
$\lim_{n\to \infty}[\frac{1}{a_{1}a_{2}}+\frac{1}{a_{2}a_{3}}\cdot\cdot\cdot\frac{1}{a_{n-1}a_{n}}].$ Let $a_{1}=1$ and $a_{n}=a_{n-1}+4$ for $n\geq 2.$ I have to find $\lim_{n\to \infty}[\frac{1}{a_{1}a_{2}}+\frac{1}{a_{2}a_{3}}\cdot\cdot\cdot\frac{1}{a_{n}a_{n-1}}].$ I tried it as
$\lim_{n\to \infty}[\frac{1}{a_{1}a... | \begin{eqnarray}E(n)=\frac{1}{a_{1}a_{2}}+\frac{1}{a_{2}a_{3}}\cdot\cdot\cdot\frac{1}{a_{n}a_{n-1}}& =&{1\over 4}\Big(\frac{a_2-a_1}{a_1a_2}+\frac{a_3-a_2}{a_{2}a_{3}}\cdot\cdot\cdot\frac{a_{n+1-a_n}}{a_{n}a_{n-1}}\Big)\\
& =&{1\over 4}\Big(\frac{1}{a_1}-\frac{1}{a_2}+\frac{1}{a_2}-\frac{1}{a_3}\cdot\cdot\cdot\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Why does multiplying by $\textbf{A}^T$ make a previously unsolvable linear system solvable Consider for instance the linear system:
$$\left(
\begin{array}{cc}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{array}
\right).\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)=\left(
\begin{array}{c}
1 \\
2 \\
4 \\
\end{array}
\... | The solutions to $Ax=b$ are the same as those of $PAx=Pb$ if $P$ is one-to-one, i.e. $\ker(P) = \{0\}$. If $P$ is not one-to-one, so that $P y = 0$ for some $y \ne 0$, then any $x$ such that $Ax = b + y$ is a solution of $PAx = Pb$ but not a solution of $Ax = b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2583454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
} |
How can I prove that $\prod\limits_{n=1}^{\infty}\left(1-\frac{1}{x_{n}^{3/2}}\right)^{-1}=\frac{3}{2}$ if $x_{n}=x_{n-1}+\sqrt{x_{n-1}}+1, x_{0}=1$? If
$$x_{n}=x_{n-1}+\sqrt{x_{n-1}}+1, x_{0}=1$$
so we can say, that
$$\prod\limits_{n=1}^{\infty}\left(1-\frac{1}{x_{n}^{3/2}}\right)^{-1}=\left[\left(1-\frac{1}{3^{3/2}}\... | Let $u_n = \sqrt{x_n}$, we have
$$u_n^2 = u_{n-1}^2 + u_{n-1} + 1 = \frac{u_{n-1}^3 - 1}{u_{n-1} - 1}
\implies u_{n-1}^3-1 = u_n^2(u_{n-1}-1)
$$
This leads to
$$1 - \frac{1}{x_n^{3/2}}
= \frac{u_n^3-1}{u_n^3}
= \frac{u_{n+1}^2(u_n-1)}{u_n^3}
= \frac{u_{n+1}^2(u_n^2-1)}{u_n^3(u_n+1)}
= \frac{u_{n+1}^2(u_n^2-1)}{u_n^2(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to calculate a combinatorial sum by hand My question is if there is an "easy" way to calculate
$$
\displaystyle \sum_{k=0}^{6}(2k)\binom{6}{k}(2)^k(3)^{6-k}
$$
I know that if it were just $\displaystyle \sum_{k=0}^{6}\binom{6}{k}(2)^k(3)^{6-k}$, this is the same as $(2+3)^6 = 5^6 = 15625$. But with that $2n$ term i... | You can write this as
\begin{align}
\sum_{k=0}^{6}(2k)\binom{6}{k}(2)^k(3)^{6-k} &= 2\cdot3^6 \sum_{k=0}^{6}k\binom{6}{k}(2)^k(3)^{-k} \\
&= 2\cdot3^6 \sum_{k=0}^{6}k\binom{6}{k} \left(\frac{2}{3}\right)^k, \\
\end{align}
and use the identity
$$\sum_{k = 0}^n k \binom{n}{k} x^k = x \frac{d}{dx} (1 + x)^n = nx(1+x)^{n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2590615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Distribution of $X+\frac{2}{X}$ when $X\sim\mathcal U(1,2)$ I have the following question:
If $X$ is a continuous random variable that is uniformly distributed on the interval $(1,2)$ what is the distribution function of $Y=X+\frac{2}{X}?$\
I have tried to calculate the inverse of the function $f(x)=x+\frac{2}{x}$ but ... | The function $x\mapsto x + \frac 2 x$ has a minimum point at $x=\sqrt 2,$ which is within the interval $(1,2),$ so it doesn't have an inverse on that interval. That makes things a bit more complicated. One thing that will make things simpler is that when $x={}$either $1$ or $2$ then $x+ \frac 2 x=3,$ i.e. it's the same... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2593442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find the range of $f(x)=11\cos^2x+3\sin^2x+6\sin x\cos x+5$ I'm trying to solve this problem.
Find the range of $f(x)=11\cos^2x+3\sin^2x+6\sin x\cos x+5$
I have simplified this problem to $$f(x)= 8\cos^2x+6\sin x\cos x+8$$ and tried working with $g(x)= 8\cos^2x+6\sin x\cos x$.
I factored out the $2\cos x$ and rewrote... | $$f(x)=8\left(\frac{1+\cos {2x}}{2}\right) +3\sin {2x}+8$$
$$f(x)=4\cos {2x}+3\sin {2x} +12$$
The range of $$4\cos {2x}+3\sin {2x}$$ is $[-5,5]$
Hence maximum and minimum values of expression are $17$ and $7$ respectively.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Prove that $\sum^m_{k=0}\binom{m}{k}(-1)^k(\frac{1}{n+k+1}) = \frac{m!n!}{(m+n+1)!}$ I am looking for a more direct proof of the following identity: $$\sum^m_{k=0}\frac{\binom{m}{k}(-1)^k}{n+k+1} = \frac{m!n!}{(m+n+1)!}$$
My original proof comes from evaluating $\int^1_0{x^n(1-x)^m}{dx}$, $n, m \in \mathbb{N}_0$ in two... | Induction on $m$.
The $m=0$ case is trivial.
For $m>0$,
$$
\begin{align}
\sum_{k=0}^m \frac{\binom{m}{k}(-1)^k}{n+k+1} &=
\sum_{k=1}^m \frac{\binom{m-1}{k-1}(-1)^k}{n+k+1} + \sum_{k=0}^{m-1} \frac{\binom{m-1}{k}(-1)^k}{n+k+1} \\ &=
\sum_{j=0}^{m-1} \frac{\binom{m-1}{j}(-1)^{j+1}}{(n+1)+j+1} + \sum_{k=0}^{m-1} \frac{\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Geometric proof for $\tan{(3x)}=\tan{(x)}\tan{\left(\frac{\pi}{3}-x\right)}\tan{\left(\frac{\pi}{3}+x\right)}$ Are there geometric proofs for the identitity
$$\tan{(3x)}=\tan{(x)}\tan{\left(\frac{\pi}{3}-x\right)}\tan{\left(\frac{\pi}{3}+x\right)}$$
My try:
Thank in advances.
|
$A,B,C,D,E$ are the same points in the problem.
$F= AB \cap CD$.
$G$ and $C$ are symmetric with respect to $E$. $H$ and $B$ are symmetric with respect to $E$.
$P = GD \cap AF$, $Q = AH \cap DF$. $R = GD \cap AH$.
(In my figure, I forgot to write $R$. Sorry.)
We want to show that $R$ is the incenter of $\triangle AD... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2596954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Difference of values of continuous function $f:\left[\frac{1}{2 \sqrt{2}},2\sqrt{2}\right]\to\mathbf{R}$ is continuous and $f\left(2\sqrt{2}\right)-f\left(\frac{1}{2 \sqrt{2}}\right)=3$. How do I show that for some $x$ in the domain $f(2x)-f(x)=1$?
| Define $g(x) = f(2x) - f(x)$ for $\displaystyle x \in \left[\frac{1}{2\sqrt{2}}, \sqrt{2}\right]$. Note that $\displaystyle \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4}$, thus$$
g\left(\frac{\sqrt{2}}{4}\right) + g\left(\frac{\sqrt{2}}{2}\right) + g(\sqrt{2}) = f(2\sqrt{2}) - f\left(\frac{\sqrt{2}}{4}\right) = 3.
$$
If any... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Average value of complex valued function property If $f:[a,b] \to \mathbb{C}$ is a continuous function define the average of $f$
$$A = \frac{1}{b-a} \int_a^b f(x)dx$$
where $\int f= \text{Re} (f) + i \int \text{Im}(f)$
Then I need to show that if $|f| \le |A|$ on $[a,b]$ then $f=A$. How to approach this?
| Assume that $A\ne 0$. We have
\begin{align*}
|A|&=\dfrac{1}{b-a}\left|\int_{a}^{b}f(x)dx\right|\leq\dfrac{1}{b-a}\int_{a}^{b}|f(x)|dx\leq\dfrac{1}{b-a}\int_{a}^{b}|A|dx=|A|,
\end{align*}
so
\begin{align*}
\dfrac{1}{b-a}\int_{a}^{b}(|A|-|f(x)|)dx=0,
\end{align*}
but $|A|-|f(\cdot)|\geq 0$ and $|A|-|f|$ is continuous, o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2607029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\frac{1}{\sqrt{ab+a+2}}+ \frac{1}{\sqrt{bc+b+2}}+ \frac{1}{\sqrt{ac+c+2}} \leq \frac{3}{2}$ Let $abc=1$ and $a,b,c>0$.
Prove that
$$\frac {1}{\sqrt {ab+a+2}}+ \frac {1}{\sqrt {bc+b+2}}+ \frac {1}{\sqrt {ac+c+2}} \leq \frac {3}{2}.$$
I guess that $$\frac {1}{\sqrt {ab+a+2}}+ \frac {1}{\sqrt {bc+b+2}}+ \frac... | $\mathbf {Hint: }$
Let $$\frac{1}{\sqrt{ab+a+2}}+ \frac{1}{\sqrt{bc+b+2}}+ \frac{1}{\sqrt{ac+c+2}}=A$$
Applying C-S:
$$\left(\frac {1}{ab+a+2}+\frac {1}{bc+b+2}+\frac {1}{ac+c+2}\right)(3) \ge (A)^2$$
Hence $$A\le \sqrt{\left(\frac {1}{ab+a+2}+\frac {1}{bc+b+2}+\frac {1}{ac+c+2}\right)(3)}$$
Now consider by Titu's lem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2608726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Generalising a question about divisibility of two numbers raised to a power Let $x$ and $y$ be integers greater than or equal to two. It is fairly straightforward to prove that if $x^3|y^2$ then $x|y$. I am interested in a more general question: when does $x^n| y^m$ imply that $x|y$ for $n,m\in\mathbb{N}$ with $n\geq m... | Simple enough.
Let $x = \prod p_i^{k_i}$ be the prime factorisation of $x$.
Then $x^m|y^n$ means $p_i^{m\cdot k_i}|y^n$ so $p_i|y$ for each prime factor $p_i$ of $x$..
Let $l_i$ be "the power of $p_i$ in $y$". i.e. Let $p_i^{l_i}|y$ so that $p_i^{l_i + 1} \not \mid y$.
Then $p_i^{m\cdot k_i}|p_i^{n\cdot l_i}$ so $m\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2609604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How can I prove that $7+7^2+7^3+...+7^{4n} = 100*a$ (while a is natural number)? How can I prove that $7+7^2+7^3+...+7^{4n} = 100*a$ (while a is entire number) ?
I thought to calculate $S_{4n}$ according to:
$$ S_{4n} = \frac{7(7^{4n}-1)}{7-1} = \frac{7(7^{4n}-1)}{6} $$
But know, I don't know how to continue for ge... | Using the Formula for a Geometric Series
We need to show more than $7^4\equiv1\pmod{100}$. If that were all we knew, then because $2\mid6$, all we would know would be
$$
\frac{7\left(7^{4n}-1\right)}{6}\equiv0\pmod{50}
$$
However, since $7^4=2401\equiv1\pmod{800}$, we know that
$$
\frac{7\left(7^{4n}-1\right)}{6}\equiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
Gradient descent proof: justify $\left(\dfrac{\kappa - 1}{\kappa + 1}\right)^2 \leq \exp(-\dfrac{4t}{\kappa+1})$ A claim on pg 279 of the notes states that:
How can the encircled be justified? Note that $\kappa := \beta /\alpha$ is a non-negative constant.
I tried using the definition of the exponential:
$$e^z = \lim\... | You overlooked that in the last inequality the iterate changes to $1$ from $t$. So the inequality you need is just
\begin{align*}
\left(\left(1 - \frac {2} {\kappa + 1} \right)^2 \right)^t \le \left(e^{ \frac{-4} {\kappa +1} } \right)^t.
\end{align*}
For the remaining part you only need to note
\begin{align*}
\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2611646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Does $x+\sqrt{x}$ ever round to a perfect square, given $x\in \mathbb{N}$? I'll define rounding as $$R(x)=\begin{cases} \lfloor x \rfloor, & x-\lfloor x \rfloor <0.5 \\ \lceil x \rceil, & else\end{cases}$$
Does $x+\sqrt{x}$ ever round (to the nearest integer) to a perfect square, given $x\in \mathbb{N}$?
For example... | $$\sqrt{n^2+n+1}= \sqrt{\left(n+\frac12\right)^2+\frac 34} > n+\frac 12$$
And $$\sqrt{n^2+n+1} < \sqrt{n^2+2n+1} =n+1$$
$$\implies n+0.5 <\sqrt{n^2+n+1} <n+1$$
Thus $\rm{fractional part}{(n^2+n+1)}>0.5$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 1
} |
Evaluating: $\int \frac {1+\sin (x)}{1+\cos (x)} dx$
Evaluate: $\int \dfrac {1+\sin (x)}{1+\cos (x)} dx$
My Attempt:
$$=\int \dfrac {1+\sin (x)}{1+\cos (x)} dx$$
$$=\int \dfrac {(\sin (\dfrac {x}{2}) + \cos (\dfrac {x}{2}))^2}{2\cos^2 (\dfrac {x}{2})} dx$$
$$=\dfrac {1}{2} \int (\dfrac {\sin (\dfrac {x}{2}) + \cos (\... | *
*OP's question: how to continue?
*existing answers: start over with simpler steps
So I feel the need to respond to the original question.
\begin{align}
\int \frac {1+\sin (x)}{1+\cos (x)} dx
&=\frac {1}{2} \int (\tan (\frac {x}{2}) +1)^2 dx \\
&= \frac12 \int (\tan^2(\frac x2)+1) dx + \int \tan(\frac x2) dx \\
&=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 3
} |
Related to matrix Find the values of $a$ and $b$ for which the equations are consistent
$$\begin{cases}
X+aY+Z=3 \\
X+2Y+2Z =b \\
X+5Y+3Z=9 \\
\end{cases}$$
I tried solving it although my phone is not uploading the snap of attempt I made. But I get to a point where I have to apply
$R_3-(R_2+\lambda)$
Is this valid?
| Consider equations$$(1)\qquad x+ay+z=3\\(2)\qquad x+2y+2z=b\\(3)\qquad x+5y+3z=9$$$(2)$ and $(3)$ are obviously linearly independent. Here we have two different cases:
Case 1: $(1)$ is independent from $(2)$ and $(3)$
At this case the determinant of $\begin{pmatrix} 1 & a &1 \\ 1 & 2 &2 \\1 & 5 &3 \\\end{pmatrix}$ is n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2621754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Partial Fraction problem solution deviates from the Rule Question:
Compute $\displaystyle \int\frac{x^2+1}{(x^2+2)(x+1)} \, dx$
My Approach:
As per my knowledge this integral can be divided in partial Fraction of form $\dfrac{Ax+B}{x^2+px+q}$ and then do the following as per to integrate it.
Solution:
Taking $\dfrac{x^... | Dividing $x^2+1$ by $x^2+2$ yields $1$ as the quotient and $-1$ as the remainder, so we have
$$
\frac{x^2+1}{x^2+2} = 1 - \frac 1 {x^2+2}.
$$
So
\begin{align}
& \frac{x^2+1}{(x^2+2)(x+1)} = \frac 1 {x+1} - \frac 1 {(x^2+2)(x+1)} \\[15pt]
= {} & \frac 1 {x+1} + \frac{Ax+B}{x^2+2} + \frac{\text{some constant}}{x+1} \\[15... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2622263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
System of ODE with non constant coefficients I have this system
$$\begin{bmatrix}\frac{dy_1(x)}{dx} \\ \frac{dy_2(x)}{dx} \end{bmatrix}=\begin{bmatrix}
\frac{1}{x} & \frac{1}{x} \\
\frac{4}{x} & \frac{1}{x}
\end{bmatrix}\begin{bmatrix}y_1(x) \\ y_2(x) \end{bmatrix}$$
I can solve system of ODE with constant... | Let $A=\begin{bmatrix}1 & 1 \\ 4 & 1\end{bmatrix}$, $y=\begin{bmatrix}y_1 \\ y_2\end{bmatrix}$ and write the equation as
$$
\frac{dy}{dx}-\frac{1}{x}Ay = 0.
$$
The characteristic polynomial of $A$ is $(\lambda-1)^2-4=(\lambda-3)(\lambda+1)$. So $A$ diagonalizable. It has a basis of eigenvectors
$$
X_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve the equation $(1-x)\sqrt{1-x^4}=x$
Solve the equation
$$(1-x)\sqrt{1-x^4}=x$$
My work so far:
$$(1-x)^2(1-x^4)=x^2$$
$$(1-2x+x^2)(1-x^4)=x^2$$
$$1-x^4-2x+2x^5+x^2-x^6=x^2$$
| You are completely correct, we obtain the polynomial equation
$$
x^6 - 2x^5 + x^4 + 2x - 1=0.
$$
The polynomial is irreducible over $\mathbb{Q}$, i.e., it does not factor into factors of smaller degree. We have exactly two real roots of the degree $6$ polynomial equation, namely
$x=0.492425875905$ and $x=- 0.9356356305... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2625156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why $ (n+1)(1-x)c^n \rightarrow_{n \rightarrow +\infty} 0$ Simple question.
Why the function $f_n(x) = (n+1)(1-x)c^n \rightarrow_{n \rightarrow +\infty} 0$ if $|c| < 1$.
I know that $(1-x)c^n \rightarrow 0$ but $(n+1) \rightarrow + \infty$. Why $f_n(x) \rightarrow 0$ ?
| $a_n =( n+1)c^n$, where $|c| < 1.$
$|a_n| = (n+1)(|c|^n)$ .
Set $b:=1/|c|$, where $b >1$.
$b = 1+a$, $a>0$.
$b^n =(1+a)^n =$
$1+ na +n(n-1)/2! + ...$
Hence:
$|a_n| = \dfrac{n+1}{b^n}=$
$\dfrac{n+1}{1+na+(n(n-1)/2!)a^2 +...} \lt$
$\dfrac{2(n+1)}{n(n-1)a^2}=$
$\dfrac{2}{(n-1)a^2} +\dfrac{2}{n(n-1)a^2}.$
The limit $n \rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Minimizing expression under constraint I was asked to minimize the expression $x^2+2y^2+3z^2$ under the constraint $xy+yz+zx=1$.
Using the Lagrange-multipliers method, the system to solve, to get eventual extrema, is
$$ \begin{cases} 2x = \lambda(y+z) \\ 4y=\lambda(x+z) \\ 6z=\lambda(x+y) \\ xy + yz + xz = 1\end{case... | We need to find a maximal value of $k$, for which the inequality
$$x^2+2y^2+3z^2\geq k(xy+xz+yz)$$ is true for all reals $x$, $y$ and $z$ or
$$3z^2-k(x+y)z+x^2+2y^2-kxy\geq0,$$ for which we need
$$k^2(x+y)^2-12(x^2+2y^2-kxy)\leq0$$ or
$$(12-k^2)x^2-2(k^2+6k)xy+(24-k^2)y^2\geq0,$$ for which we need $12-k^2>0$ and
$$(x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Beckenbach Introduction to Inequalities Chapter 2: Show $(a+b)/2 \le ( (a^2 + b^2 )/2)^{1/2}$ I'm having trouble understanding the following problem
Problem
Beckenbach, Chapter 2 Pg 24 Ex 1
$$
\text{Show the following for all a, b}\quad
\frac{(a+b)}{2} \le \left(\frac{a^2 + b^2}{2}\right)^\frac{1}{2}
$$
The book pr... | *
*Your proof has a tiny issue you need to address (and then it'll be correct): you forgot absolute values around $\frac{a+b}{2}$ once you factor $\left(\frac{a+b}{2}\right)^2$ out of the square root.
*Note that
$$
\frac{a+b}{2} \leq \frac{\lvert a\rvert +\lvert b\rvert}{2}\tag{1}
$$
so it suffices to prove
$$
\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Solution to first order partial Cauchy problem? Solve the Cauchy Problem :
$$(x+y)u_{x} + (x-y)u_y = 1$$ with $ \; \; u(1,y)=\frac{1}{\sqrt{2}}$
My attempt:
We have the following characteristic equations :
$$\frac{dx}{x+y} = \frac{dy}{x-y} = \frac{du}{1}$$
Solving
$$\frac{dx}{x+y} = \frac{dy}{x-y} \Rightarrow xdx - yd... | Your first characteristic equation is correct :
$$x^2-y^2-2xy=c_1$$
From this : $\quad x-y=\pm\sqrt{c_1+2y^2}$
Second characteristic, from $\quad\frac{dy}{x-y}=\frac{du}{1}=\frac{dy}{\pm\sqrt{c_1+2y^2}}$
$u-\int\frac{dy}{\pm\sqrt{c_1+2y^2}}=c_2$
$u\pm\frac{1}{\sqrt{2}}\ln\bigg|\pm\sqrt{2(c_1+2y^2)}+2y\bigg|=c_2$
$u\pm\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2631374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to prove $\tan\Big[\frac{1}{2}\sin^{-1}\frac{3}{4}\Big]=\frac{4-\sqrt{7}}{3}$ Prove
$$
\tan\Big[\frac{1}{2}\sin^{-1}\frac{3}{4}\Big]=\frac{4-\sqrt{7}}{3}
$$
and justify why $\frac{4+\sqrt{7}}{3}$ is ignored.
My Attempt:
$$
\tan x=\frac{2\tan\frac{x}{2}}{1-\tan^2\frac{x}{2}}\implies\tan x-\tan x\tan^2\frac{x}{2}... | $$\tan\left[\frac{1}{2}\sin^{-1} \left(\frac{3}{4}\right)\right]=\frac{4-\sqrt{7}}{3}$$
$\sin^{-1}(x) = \arcsin(x) = 2\arctan \left(\frac x{1+\sqrt{1-x^2}}\right)$
See Inverse trigonometric functions
\begin{align}
\tan \left[\frac{1}{2}\sin^{-1} \left(\frac{3}{4}\right)\right] &=\tan \left[\frac{1}{2}2\arctan \left(\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2632633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Some implicit differentiation questions. Please check them [see desc.] I have a few implicit diffentiatial questions that I wanted to check. general question... how do you know that $y$ is a function of $x$? I assume the whole reason why these are called implicit differentiation questions is because $y$ is defined impl... | Note that in the case of implicit differentiation it is not necessarily the case that one variable is a function of the other in the strict sense, but the two functions are related. One can symbolize this as $x$ and $y$ being parametric functions of a variable $t$. An alternate approach to implicit differentiation is b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2637866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding A Matrix Representation
Let $T=\begin{pmatrix}
3x+3y+3z \\
4x+7y+4z\\
6x+8y+6z
\end{pmatrix}$ And basis $B=\left\{\begin{pmatrix}
3 \\
1\\
2
\end{pmatrix},\begin{pmatrix}
4 \\
2\\
3
\end{pmatrix},\begin{pmatrix}
2 \\
1\\
2
\end{pmatrix}\right\}$ Find $[T]_B$
Now to find it directly I applied T on th... | Note that
$$[T]_E=
\begin{pmatrix}
3 & 3 & 3 \\
4 & 7 & 4\\
6 & 8& 6
\end{pmatrix}$$
and
$$[M]_{EB}=
\begin{pmatrix}
3 & 4 & 2 \\
1 & 2 & 1\\
2 & 3& 2
\end{pmatrix}$$
$$v_E=[M]_{EB} \cdot v_B\implies v_B=[M]_{EB}^{-1}\cdot v_E \implies v_B= [M]_{BE}\cdot v_E$$
thus for $w=T(v)$ we have
$$w_E=[T]_{E}\cdot v_E\impl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove the positivity of a sequence Let $a>0$ a real number and $(u_n)$ the sequence defined by
$$
u_{n+1} = a - \frac{1}{u_n}\text{ and } u_0 = a.
$$
Question: Determine condition on the value of $a>0$ such that the sequence $(u_n)$ is always positive.
Attempt: I tried to establish a general formula of $u_n$ in order... | hint
Let $f (x)=a-1/x $.
$f $ is increasing at $(0,+\infty) $.
$(u_n) $ is monotonic.
$u_1 <u_0$ thus it is strictly decreasing.
the sequences terms are $>0$ if the limit (if it exists) is $\ge 0$.
the limit $l $ satisfies $l=a-1/l .$
if $a\ge 2$ then
$l=(a\pm\sqrt {a^2-4})/2>0$.
If $a<2$ , $u_2<0$.
So, the condition... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
PDF and CDF of Sum of 2 dice i'm trying to obtain the PDF and CDF of the sum of 2 dice toss.
There are tons of elementary exercise where is asked to find the exact probability,
but what about the PDF and CDF?
i thought this is a convolution of discrete uniform PDF. but i don't know where to start to find it.
it's like ... | How many ways are there to get a sum of $2$? We must get $1$ and then $1$ again with probability $\frac{1}{36}$.
How many ways are there to get a sum of $3$? We must either get a $1$ and then $2$ or vice versa with probability $\frac{2}{36}$.
How many ways are there to get a sum of $4$? We must either get a $1$ and th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2639380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Sum of the series $\csc^{-1} \sqrt{10}+ \csc^{-1} \sqrt{50}+\csc^{-1}\sqrt{170}...$
Find the sum of the series:
$\csc^{-1}\sqrt{10}+ \csc^{-1}\sqrt{50}+\csc^{-1}\sqrt{170}...\csc^{-1}\sqrt{(n^2+1)(n^2+2n+2)}$
I converted the series to $\sum^{n} _{i=0}\arcsin \dfrac{1}{\sqrt{(i^2+1)(i^2+2i+2)}}$ and then tried to u... | Use $$\arcsin\frac{1}{\sqrt{n^2+1}}-\arcsin\frac{1}{\sqrt{n^2+2n+2}}=\arcsin\frac{1}{\sqrt{(n^2+1)(n^2+2n+2)}}$$ and the telescopic sum.
Indeed,
$$\sin\left(\arcsin\frac{1}{\sqrt{n^2+1}}-\arcsin\frac{1}{\sqrt{n^2+2n+2}}\right)=$$
$$=\frac{1}{\sqrt{n^2+1}}\cdot\frac{n+1}{\sqrt{n^2+2n+2}}-\frac{n}{\sqrt{n^2+1}}\cdot\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2644191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to find the value of a and b in limit if the denominator = 0? Given that
$$
\lim_{x \rightarrow 3}\frac{4x^3-bx^2+2x+30}{x^3-ax+3a-27} = \frac{1}{2}
$$
Find the value of a and b.
I have using
$$
\lim_{x \rightarrow 3}4x^3-bx^2+2x+30 = 0
$$
To found that b = 16
Does this mean that
$$
\lim_{x \rightarrow 3}\frac{4... | You correctly realise that it must be a $0/0$ form since denominator goes zero for the limit, so numerator and denominator must have a common root. Since denominator has a root $x = 3$, numerator must also be divisible by $x-3$.
But as Hagan says, you don't know the multiplicity of the root $x=3$ in numerator or denom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2645935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Calculate the determinant $\left|\begin{smallmatrix} a&b&c&d\\ b&a&d&c\\ c&d&a&b\\d&c&b&a\end{smallmatrix}\right|$ Question: Calculate the following determinant
$$A=\begin{vmatrix} a&b&c&d\\ b&a&d&c\\ c&d&a&b\\d&c&b&a\end{vmatrix}$$
Progress: So I apply $R1'=R1+R2+R3+R4$ and get
$$A=(a+b+c+d)\begin{vmatrix} 1&1&1&1\\ b... | This is a block matrix, you can calculate a determinant of a $2 \times 2$ matrix made of $2 \times 2$ matrices, and then expand that: $$A=\begin{vmatrix} M&N\\ N&M\end{vmatrix}$$ where $$M=\begin{bmatrix} a&b\\ b&a\end{bmatrix}$$ and $$N=\begin{bmatrix} c&d\\ d&c\end{bmatrix}$$
All matrices are symmetric, this is reall... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2647193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
A sum of series problem: $\frac{3}{1!+2!+3!} + \frac{4}{2!+3!+4!} + \cdots + \frac{2008}{2006!+2007!+2008!}$ I have a question regarding the sum of this series:
$$\frac{3}{1!+2!+3!} + \frac{4}{2!+3!+4!} + \cdots + \frac{2008}{2006!+2007!+2008!}$$
My approach:
I found that this sum is equal to:
$$\sum_{n=3}^{2008}\frac{... | $\sum_{n=3}^{2008}\frac{1}{n(n-2)!}=\sum_{n=3}^{2008}\frac{1}{n(n-2)!}.\frac{(n-1)}{(n-1)}=\sum_{n=3}^{2008}\frac{n-1}{n!}=\sum_{n=3}^{2008}(\frac{n}{n!}-\frac{1}{n!})=\sum_{n=3}^{2008}(\frac{1}{(n-1)!}-\frac{1}{n!})=\frac{1}{2!}-\frac{1}{3!}+\frac{1}{3!}-\frac{1}{4!}+...-\frac{1}{2007!}+\frac{1}{2007!}-\frac{1}{2008!}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2647385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 3
} |
number of distinct terms in binomial expansion of $\left(x+x^{-1}+x^2+x^{-2}\right)^{15}$ Question -> How do I find the number of distinct terms in the binomial expansion of $$\left(x+\frac{1}{x}+x^2+\frac{1}{x^2}\right)^{15}$$
Solution I tried ->
$$\left(\frac{x^3+x+x^4+1}{x^{2}}\right)^{15}=x^{-30}\left(1+x+x^3+x^4\r... | We can get any exponent from $0$ to $60$. Thus, there are $61$ distinct terms.
Consider
$$
\left(1+x+x^3+x^4\right)^{15}
$$
For $0\le k\le30$, there is a way to get $x^k$ by
$$
\color{#C00}{1}^{15-k+2\lfloor k/3\rfloor}\color{#C00}{x}^{k-3\lfloor k/3\rfloor}\color{#C00}{x}^{\color{#C00}{3}\lfloor k/3\rfloor}
$$
since $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2648294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
How to prove that $x^2yz+xy^2z+xyz^2 \leq \frac{1}{3}$ where $x^2+y^2+z^2 = 1$ and $x,y,z >0$? How to prove that $x^2yz+xy^2z+xyz^2 \leq \frac{1}{3}$ where $x^2+y^2+z^2 = 1$ and $x,y,z >0$?
| By Cauchy–Schwarz inequality with
*
*$u=(x,y,z)$
*$v=(xyz,xyz,xyz)$
we have
$$u\cdot v\le |u|\cdot |v|$$
that is
$$x^2yz+xy^2z+xyz^2 \leq\sqrt3\,xyz \le \frac{1}{3}$$
indeed by AM-GM
$$\frac13=\frac{x^2+y^2+z^2}{3}\ge \sqrt[3]{x^2y^2z^2}\implies xyz \le\frac{1}{3\sqrt3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2648962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Powers of ten addition with negative power I have been stumped by my nephew's algebra homework.
He has a question:
$10^5 + 10^{-3}=$?
The multi-choice answers were:
$10^{-5},10^{15},10^{3},10^{2},10^{5},10^{8}$
(to 1 s.f.)
I thought it was as simple as adding the powers - therefore being $10^2$. Apparently not. Can som... | I will assume that by $10^5+10^{-3}$ you meant $10^5\times 10^{-3}$.
Say we have a value $x$ and we want to raise it to a power $n$. This means that we multiply $x$ by itself $n$ times. $$x^n = \underbrace{x\cdot x\cdot x\cdot\ldots\cdot x}_{n\text{ times.}}\tag1$$ It is confusing to most people when we say that $x^0 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Let $(x,y) \in \Bbb R^+$ Prove that $\Bigr(1+\frac{1}{x}\Bigl)\Bigr(1+\frac{1}{y}\Bigl)\ge \Bigr(1+\frac{2}{x+y}\Bigl)^2$ Let $(x,y) \in \Bbb R^+$ Prove that $$\Bigr(1+\frac{1}{x}\Bigl)\Bigr(1+\frac{1}{y}\Bigl)\ge \Bigr(1+\frac{2}{x+y}\Bigl)^2$$
My try
Well, i didn't see a way to factorize this, so i put it in WolframA... | it is just $AM-GM$
multiplying out we get
$$(x+y)^2+\frac{(x+y)^2}{x}+\frac{(x+y)^2}{y}+\frac{(x+y)^2}{xy}\geq (x+y)^2+4+4(x+y)$$ and this is
$$xy(x+y)^2+y(x+y)^2+x(x+y)^2+(x+y)^2\geq xy(x+y)^2+4xy+4xy(x+y)$$
$$(x+y)^3+(x+y)^2\geq 4xy+4xy(x+y)$$
and this is $$(x+y)^2(x+y+1)\geq 4xy(x+y+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2650818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 3
} |
Find the equation of the plane which bisects the pair of planes $2x-3y+6z+2=0$ and $2x+y-2z=4$ at acute angles.
Find the equation of the plane which bisects the pair of planes $2x-3y+6z+2=0$ and $2x+y-2z=4$ at acute angles.
My try:
The normal to the plane $2x-3y+6z+2=0$ has direction ratios $2,-3,6$ and the normal t... | Let $(x,y,z)$ be a point in the needed plane.
Thus, $$\frac{|2x-3y+6z+2|}{\sqrt{2^2+(-3)^2+6^2}}=\frac{|2x+y-2z-4|}{\sqrt{2^2+1^2+(-2)^2}},$$
which gives equations of two our planes.
I got
$$x+2y-4z-\frac{17}{4}=0;$$
$$10x-y+2z-11=0.$$
Now, take $A(1,1,1)$, which is placed on the second plane.
Let $AB$ be a perpendicu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Collinearity when $\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0}$
Let $\mathbf{a} = \begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}$, $\mathbf{b} = \begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix}x_c\\y_c\\z_c\end{pmatrix}$.
Show that $... | Using that $\,\color{blue}{\mathbf{b} \times \mathbf{c} = -\,\mathbf{c} \times \mathbf{b}}\,$ and $\,\color{red}{\mathbf{a} \times \mathbf{a} = 0}\,$:
$$
\begin{align}
\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a} = \mathbf{0} \;\;&\iff\;\; \mathbf{a} \times \mathbf{b} \colo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Simplify the equation $16\cos^6(t) + 16\sin^6(t) + 48\sin^2(t)\cos^2(t)$ I am trying to simplify the equation
$$16\cos^6(t) + 16\sin^6(t) + 48\sin^2(t)\cos^2(t)$$
The goal here is to prove that this can be simplified in a scalar value.
Thanks!
| $(a^2+b^2)^3 = a^6 + 3a^4b^2 + 3a^2b^2 + b^6 \Rightarrow$
$ a^6+b^6 = (a^2+b^2)^3 - 3a^2b^2(a^2+b^2)$
Substituting $a=\sin(x)$ and $b=\cos(x)$:
$ \sin^6(x)+\cos^6(x) = (\sin^2(x)+\cos^2(x))^3 - 3\sin^2(x)\cos^2(x)(\sin^2(x)+\cos^2(x)) = \\
1 - 3\sin^2(x)\cos^2(x) \Rightarrow$
$ 16\sin^6(x)+16\cos^6(x) = 16 - 48\sin^2(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Determine what condition the $n\times n$ matrix $A$ must fulfill so that the matrix $I_n - A$ has an inverse and this equals $I_n + A$.
Determine what condition the $n\times n$ matrix $A$ must fulfill so that the matrix $I_n - A$ has an inverse and this equals $I_n + A$.
I tried to apply it to a $2\times 2$ matrix ob... | $(I - A)^{-1} = I + A \Longleftrightarrow A^2 = 0; \tag 1$
for if
$A^2 = 0, \tag 2$
then
$(I - A)(I + A) = I + A - A - A^2 = I - A^2 = I; \tag 3$
likewise if
$(I - A)(I + A) = I, \tag 4$
then
$I - A^2 = (I - A)(I + A) = I, \tag 5$
then
$I = A^2 + I, \tag 6$
or
$A^2 = 0. \tag 7$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Block Matrix Inversion in Wikipedia Wikipedia provides two formulas for block-matrix inversion:
$$ {\begin{bmatrix}\mathbf {A} &\mathbf {B} \\\mathbf {C} &\mathbf {D} \end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} ^{-1}+\mathbf {A} ^{-1}\mathbf {B} (\mathbf {D} -\mathbf {CA} ^{-1}\mathbf {B} )^{-1}\mathbf {CA} ^{-1}&-... | Assuming $A,B, C, D$ are fixed.
By uniqueness of the inverse, yes, those expression are equal provided those terms involved indeed exists.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to extract $(x+y+z)$ or $xyz$ from the determinant Prove
$$\color{blue}{
\Delta=\begin{vmatrix}
(y+z)^2&xy&zx\\
xy&(x+z)^2&yz\\
xz&yz&(x+y)^2
\end{vmatrix}=2xyz(x+y+z)^3}
$$
using elementary operations and the properties of the determinants without expanding.
My Attempt
$$
\Delta\stackrel{C_1\rightarrow C_1+C_2+C_3... | $$
\Delta=\begin{vmatrix}
(y+z)^2&xy&zx\\
xy&(x+z)^2&yz\\
xz&yz&(x+y)^2
\end{vmatrix}=xyz\begin{vmatrix}
\frac{(y+z)^2}{x}&x&x\\
y&\frac{(x+z)^2}{y}&y\\
z&z&\frac{(x+y)^2}{z}
\end{vmatrix}=
xyz\begin{vmatrix}
\frac{(y+z)^2}{x}-x&x&0\\
y-\frac{(x+z)^2}{y}&\frac{(x+z)^2}{y}&y-\frac{(x+z)^2}{y}\\
0&z&\frac{(x+y)^2}{z}-z
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2659331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How many ways are there to get a sum of $25$ when $10$ distinct dice are rolled? I have come across the following problem and have given it a good attempt below. I am wondering if I have proceeded correctly, and if not if someone could show me the correct answer or maybe a more efficient solution, thanks!
How many way... | Can't comment yet so adding here: I haven't checked the math, but just for fun https://ideone.com/IKt8OR
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2660742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why does this innovative method of subtraction from a third grader always work? My daughter is in year $3$ and she is now working on subtraction up to $1000.$ She came up with a way of solving her simple sums that we (her parents) and her teachers can't understand.
Here is an example: $61-17$
Instead of borrowing, maki... | My way of explaining it:
\begin{align*}
61−17 &= (6 \cdot 10 + 1 \cdot 1) - (1 \cdot 10 + 7 \cdot 1) \\
&= (6 - 1) \cdot 10 + (1-7) \cdot 1 \\
&= (6 - 1) \cdot 10 - (7-1) \cdot 1 \\
&= 5 \cdot 10 - 6 \cdot 1 \\
&= 50 - 6 = 44 .
\end{align*}
On the one hand, you could generalise:
\begin{align*}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2667980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "297",
"answer_count": 18,
"answer_id": 16
} |
Proving $\frac{a}{a^2-a+1}+\frac{b}{b^2-b+1}+\frac{c}{c^2-c+1}+\frac{d}{d^2-d+1}\le \frac{8}{3}$ given $a+b+c+d=2$
Let $a,b,c,d\in \mathbb{R}$ and $a+b+c+d=2$. Prove that $$\frac{a}{a^2-a+1}+\frac{b}{b^2-b+1}+\frac{c}{c^2-c+1}+\frac{d}{d^2-d+1}\le \frac{8}{3}.$$
We have $$\frac{a}{a^2-a+1}\le \frac{4}{3}a\Longleftrig... | First, we simplify a bit by transforming $x = a-\frac12, y = b - \frac12, z = c - \frac12, w = d - \frac12$, so that we need to prove equivalently with $x+y+z+w=0$, the inequality
$$\sum \frac{2x+1}{4x^2+3} \leqslant \frac43$$
Here $\sum$ represents cyclic sums. Also as $\displaystyle 1 - 2\cdot\frac{2x+1}{4x^2+3} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2668871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proof Verification: Prove $f(n)<2$, when $f(1)=1$ and $f(n+1) = \sqrt{(2+f(n))}$ for all $n$ positive integers Proceed by induction:
Suppose $P(n)$ is the statement $f(n)<2$, when $n$ $f(1)=1$ and $f(n+1) = \sqrt{(2+f(n))}$ for all $n$ positive integers.
Base case: $P(1)=f(1)=1<2$. True.
Assume $P(n)$, i.e. $f(n)<2$ wh... | Alt. hint: obviously $f(n) \ge 0$ for all $\,n\,$, then:
$$
\begin{align}
f(n+1) = \sqrt{f(n)+2} \quad&\iff\quad \left(f(n+1)\right)^2 = f(n)+2 \\
&\iff\quad \left(f(n+1)\right)^2 -4 = f(n) - 2 \\
&\iff\quad \big(f(n+1)-2\big)\underbrace{\big(f(n+1)+2\big)}_{\gt 0}=f(n)-2
\end{align}
$$
It follows that all differenc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Let $a^2<2, b=2(a+1)/(a+2)$. Show $b^2<2$ (assignment) It is a part of my assignment.
$$ \text {Let }a^2<2, \quad b=2\frac {(a+1)}{(a+2)}\quad \text{ Show } b^2<2$$
I already proved that a
But, I am struggling to prove $b^2<2$.
My lecturer said that I need to manipulate $b^2$, which is larger than $b^2$ but less th... | Hint: $\require{cancel}\;b^2 - 2 = \dfrac{4(a+1)^2}{(a+2)^2}-2=\dfrac{4a^2+\cancel{8a}+4-2a^2-\cancel{8a}-8}{(a+2)^2}=\dfrac{2a^2-4}{(a+2)^2}=\dfrac{2(a^2-2)}{(a+2)^2}\,$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671699",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Find the base of the kernel of $L(y)=x^2y''-3xy'+3y.$
Let $L:C^2(I)\rightarrow C(I), L(y)=x^2y''-3xy'+3y.$ Find the kernel of the linear transformation $L$. Can the solution of $L(y)=6$ be expressed in the form $y_H$+$y_L$, where $y_H$ is an arbitrary linear combination of the elements of ker L.
What I have tried:
Si... | Another way
$$x^2y''-3xy'+3y=0$$
For $x \neq 0$
$$y''-3 \left (\frac {xy'-y}{x^2} \right )=0$$
$$y''-3\left (\frac {y}{x}\right )'=0$$
$$y'-3 \left (\frac {y}{x} \right )=K_1$$
$$x^3y'-3{y}{x^2}=K_1x^3$$
$$\frac {x^3y'-3{y}{x^2}}{x^6}=\frac {K_1}{x^3}$$
$$\left (\frac {y}{x^3} \right )'=\frac {K_1}{x^3}$$
$$\frac {y}{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2674938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solve the reccurence relation $a_n = 3a_{n-1}+4_{n-2}, a_0=a_1 = 1$ This is my first time working through this type of problem and I am looking to see if I have worked it out correctly, thanks!
Solve the reccurence relation $$a_n = 3a_{n-1}+4a_{n-2}, a_0=a_1 = 1$$
First we get the characteristic equation:
$$x^n = 3x^... | Yes it is correct. We can check it with the use of generating functions.
Transform $a_n = 3a_{n-1} + 4a_{n-2}$ to $$a_{n+2} = 3a_{n+1} + 4a_n \tag{1}$$.
Let $A(x) := \sum_{n \geq 0} a_n x^n$ then we get from (1)
$$
\begin{eqnarray*}
\sum_{n \geq 0} a_{n+2} x^n &=& 3 \sum_{n \geq 0} a_{n+1} x^n + 4 \sum_{n \geq 0} a_n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2676579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Solving a system of equations: I am having problems solving a system of equations I haven't run into a system quite like this before and I am looking for some help, thanks!
$$a(1+\sqrt3)^2+b(1-\sqrt3)^2=3$$ $$a(1+\sqrt3)^3+b(1-\sqrt3)^3=8$$
How can I solve for $a$ and $b$?
| Write it as a system $Ax=v$ with $v=(3,8)^T$, $x=(a,b)^T$ and
$$
A=\begin{pmatrix} (1+\sqrt{3})^2 & (1-\sqrt{3})^2 \cr
(1+\sqrt{3})^3 & (1-\sqrt{3})^3\end{pmatrix}
$$
Since $\det(A)=-8\sqrt{3}\neq 0$, the inverse $A^{-1}$ exists, so the unique solution is
$$
\begin{pmatrix} a \cr b \end{pmatrix}=x=A^{-1}v=\frac{1}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2676675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Symmetric Olympiad inequality Prove that for all positive reals $a, b, c$ this inequality holds:
$$\sum_{cyc}\frac{a^3}{b^2 - bc + c^2} \ge \sum_{cyc}a$$
I have proved this in a very ugly way:
I multiplied with $(a^2 - ab + b^2)(b^2 - bc + c^2)(c^2 - ca + a^2)$ and with some work i managed to prove this with Schur. I f... | Note by CS (with $\sum$ denoting cyclic sums):
$$ \sum \frac{a^3}{b^2-bc+c^2} \geqslant \frac{\left(\sum a^2 \right)^2}{\sum a(b^2-bc+c^2)}$$
Hence it is enough to show
$$\left(\sum a^2 \right)^2 \geqslant \left(\sum a\right) \cdot \sum a(b^2-bc+c^2)$$
$$\iff \sum a^4 + 2\sum a^2b^2 \geqslant \sum (2a^2b^2-a^2bc + b^3a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Integral of arctan form I am trying to solve this integral without techniques ($u$-sub/parts), just simplifying and inspection:
$$
\int \dfrac{1}{4+x^{2}}dx
$$
I notice an $\arctan$ form, but that $4$ in the denominator is confusing me, if there was a multiplying factor in front of the $x$ I would know what to do, but ... | One can take a different route with the following. Let $x = 2 t$ to obtain
\begin{align}
I &= \int \frac{dx}{4 + x^2} \\
&= \frac{1}{2} \, \int\frac{dt}{1 + t^2} = \frac{1}{4} \, \int\left(\frac{1}{1 + i t} + \frac{1}{1 - i t} \right) \, dt \\
&= \frac{1}{4} \, \left[ \frac{1}{i} \, \ln(1 + i t) - \frac{1}{i} \, \ln(1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Proving $\frac{\sqrt3\cos x-\sin x}{\sin 3x}> \frac{\sqrt3}{3x}-\frac13$ for small $x>0$
Prove that$$\frac{\sqrt3\cos x-\sin x}{\sin 3x}> \frac{\sqrt3}{3x}-\frac13$$
for small $x$ near $0$.
From Taylor expansion I can see that $$\frac{\cos x}{\sin3x}>\frac13x,\quad \frac{\sin x}{\sin3x}>\frac13,$$ but combining the... | Using the identity from @labbhattacharjee, for $0 < x < \dfrac{π}{3}$ there is\begin{align*}
0 < \frac{\sin 3x}{\sqrt{3}\cos x - \sin x} &= \frac{1}{2} \frac{\sin\left( 3\left( \dfrac{π}{3} - x \right) \right)}{\sin\left( \dfrac{π}{3} - x \right)} = \frac{1}{2} \left( 3 - 4\sin^2\left( \dfrac{π}{3} - x \right) \right)\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$x^2+y^2+z^2 = 4, z \geq 4.$
Let $\mathbf{F}=3y \ \mathbf{i} -3xz \ \mathbf{j} + (x^2-y^2) \
\mathbf{k}.$ Compute the flux of the vectorfield
$\text{curl}(\mathbf{F})$ through the semi-sphere $x^2+y^2+z^2=4, \
z\geq 0$, by using direct parameterization of the surface and computation of $\text{curl}(\mathbf{F}).$
... | Note that
*
*$(1)$ hold since $xy,xz,yz$ are antisymmetric function on the domain, indeed, for example, at each point xy in the I quadrant correspond -xy in th II and so on
*$(2)$ hold since $x^2,y^2,z^2$ are symmetric function on the domain and each single integral for $x^2,y^2,z^2$ must assume the same value for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $a+b+c<0$ and $ax^2+bx+c=0$ has no real roots, is it true that $c$ must be less than $0$? I decided to look at the graphs of the parabola to solve this problem. These are only two types that will fit this problem: parabolas that "open" upwards and parabolas that "open" downwards. These parabolas would never touch th... | By the quadratic formula a parabola has roots at $x = \frac {-b \pm \sqrt{b^2 - 4ac}}{2a}$ if such is a real number.
The only way for there not to be roots is if $b^2 - 4ac < 0$. Or if $b^2 < 4ac$.
If $c > 0$ then for this to happen we must have $0 < b^2 <4ac$ so $a > 0$ as well.
But $a+b+c < 0$ so $b$ must be negat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2684575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Finding $ a $ where $ P(x)-a $ divides with $ (x+1)^4 $ and $ P(x)+a $ divides with $ (x-1)^4 $ Let $ P(x) $ be a $ 7 $ degree polynomial with the coefficient of $ x^7 $ equal to $ 1 $. Let $ a \in\mathbb{R} $ such that $ P(x)-a $ divides with $ (x+1)^4 $ and $ P(x)+a $ divides with $ (x-1)^4 $
$ 1 ) $ Find the coeffi... | $(x+1)^4 \mid (P(x)-a) \implies P(x) = f(x)(x+1)^4+a \implies (x+1)^3 \mid P'(x)$.
Similarly we have $(x-1)^3 \mid P'(x)$, so putting these together with $P(x)$ being monic of seventh degree, $P'(x) = 7(x^2-1)^3$.
$\implies P(x) = x^7 -\frac{21}5x^5+7x^3-7x + C$,
Now $P(1) = -a, P(-1) = a \implies C-\frac{16}5=-a, C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2685002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find the largest cylinder inscribed inside a sphere. Is this calculation correct so far?
A right circular cylinder is inscribed in a sphere of radius $r$. Find the largest possible volume of such a cylinder.
I have that the radius of the cylinder is $r$, the radius of the sphere is $R$, and the height of the inscribe... | You might enjoy the fact that you actually do not need derivatives. By the AM-GM inequality
$$V^2=16\pi^2\cdot \frac{r^2}{2}\cdot \frac{r^2}{2}\cdot(R^2-r^2)\leq 16\pi^2\left(\frac{R^2}{3}\right)^3 $$
i.e. $V\leq \frac{4\pi R^3}{3\sqrt{3}}$, with equality attained at $\frac{r^2}{2}=R^2-r^2$, i.e. at $r=R\sqrt{\frac{2}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2685819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Yet another matrix equation
Solve the following matrix equation for $X$.
$$\left[\begin{array}{cc}
5 &-8\cr
8 &1
\end{array}\right] X + \left[\begin{array}{cc}
6 &6\cr
3 &5
\end{array}\right] = \left[\begin{array}{cc}
-1 &4\cr
-3 &-1
\end{array}\right] X$$
Please give me some hint to do this question. Thanks.
| This is a linear equation in four variables.
Let $X = \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \end{bmatrix}$.
Consider each entry of the RHS and LHS as a seperate equation:
$$\begin{align}
5x_{11} - 8x_{21} + 6 &= -1x_{11}+4x_{12} & \text{(top left entry)}\\
5x_{12} - 8x_{22} + 6 &= -1x_{12}+4x_{22} & \text... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2686161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
What are the maximum and minimum values of $4x + y^2$ subject to $2x^2 + y^2 = 4$? $$ 2x^2 + y^2 = 4 $$
$$ Y = \sqrt{4-2x^2} $$
$$4x + y = 2x^2 + \sqrt{4-2x^2}$$
Find the derivative of $$ 2x^2 + \sqrt{4-2x^2} $$ set as = 0
$$X^2 = 64/33$$
$$ F(64/33) = 34\sqrt{33}/33 $$
How to solve it the right way?
| Note that $y^2=-2x^2+4$
When performing substitution of $4x+y^2 \rightarrow 4x-2x^2+4$
Finding the vertex (maxima):
$$-2x^2+4x+4=-2(x^2-2x)+4=-2[(x-1)^2-1]+4=-2(x-1)^2+6$$
Which presents the relative(and only) maxima.
Note that there is no relative minima, as the parabola goes downwards.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2687820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Primes congruent to $1$ modulo $6$ are of the form $3a^2+b^2$ I am asked to prove that primes of the form $6k+1$, where $k$ is an integer can be written as $3a^2 + b^2$, where $a$ and $b$ are integers by using the fact that $\textbf Z[\omega]$ is a UFD, where $\omega^2 + \omega + 1 = 0$. However, I cannot see any conne... | If $p \equiv 1 \pmod 6$ then $\left(\frac{-3}{p}\right) = \left(\frac{-1}{p}\right)\left(\frac{3}{p}\right) = (-1)^{\frac{p-1}{2}}\left(\frac{p}{3}\right)(-1)^{\frac{p-1}{2}\frac{3-1}{2}} = \left(\frac{p}{3}\right)=1$ so $-3$ is a perfect square mod $p$.
Therefore there exists a solution to the equation $x^2 + 3 = 0$ i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
$a+b+c=?$ by having $a^2+160=b^2+5$ and $a^2+320=c^2+5$ My question:
$a+b+c=?$ by having $a^2+160=b^2+5$ and $a^2+320=c^2+5$.
My work so far:
$a^2+160=b^2+5\Rightarrow (b-a)(a+b)=155=31\times 5$
$a^2+320=c^2+5\Rightarrow (c-a)(c+a)=315=5\times3^2\times 7$
And now, I'm stuck.
($a,b,c$ are a members of $\mathbb Z$ and ar... | Hint:
$$(b+a)(b-a)=31\times 5$$
Since $a,b$ are integers, and $5,31$ are prime numbers, what is the value of $(b+a)$ and $(b-a)$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Solve $\frac{x^2}{\left(a+\sqrt{a^2+x^2}\right)^2}+\frac{x^2(1+x^2)}{\left(a+\sqrt{a^2+x^2(1+x^2)} \right)^2}=1$ I have been trying to solve the following equation for $x$:
\begin{align}
\frac{x^2}{\left(a+\sqrt{a^2+x^2}\right)^2}+\frac{x^2(1+x^2)}{\left(a+\sqrt{a^2+x^2(1+x^2)} \right)^2}=1,
\end{align}
for some fixed ... | Your equation is simplified to:
$$\dfrac{t+\sqrt{1+t^2+\frac{a^4}{t^4}}}{2t} = (t+\sqrt{1+t^2})^2$$
with the substitution $a=xt.$ Conceivably, now you can explicitly find $a$ in terms of $t$ and then might succeed doing an asymptotic analysis, but I really doubt anything close to a closed form solution exists, despite ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $ \lim_{n\to \infty} ( \lim_{x\to0} (1+\tan^2(x)+\tan^2(2x)+ \cdots + \tan^2(nx)))^{\frac{1}{n^3x^2}} $ $ \lim_{n\to \infty} ( \lim_{x\to0} (1+\tan^2(x)+\tan^2(2x)+ \cdots + \tan^2(nx)))^{\frac{1}{n^3x^2}} $
The answer should be $ {e}^\frac{1}{3} $
I haven't encountered problems like this before and I'm pretty... | Note that for $x\to 0$
*
*$\tan x= x+o(x^2)$
then
$$(1+\tan^2(x)+\tan^2(2x)+ \cdots + \tan^2(nx)))^{\frac{1}{n^3x^2}} = (1+x^2+4x^2+...+n^2x^2+o(x^2)) ^ {\frac{1}{n^3x^2}}=\large{e^{\frac{\log(1+x^2+4x^2+...+n^2x^2+o(x^2))}{n^3x^2}}=e^{\frac{x^2+4x^2+...+n^2x^2+o(x^2)}{n^3x^2}}=e^{\frac{n(2n+1)(n+1)}{6n^3}+o(1)} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2696097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find all the rational values of $x$ at which $y=\sqrt{x^2+x+3}$ is a rational number Question
Find all the rational values of $x$ at which $y=\sqrt{x^2+x+3}$
My attempt
Since we only have to find the rational values of $x$ and $y$, we can assume that
$$ x \in Q$$
$$ y \in Q$$
$$ y-x \in Q $$
Let$$ d = y-x$$
$$d=\sqrt{... | Hint:
The expression is of the following form
$$(a+b-c)^2 = a^2+b^2+c^2+2ab-2bc-2ac$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2698555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Show that $\frac{1}{3}<\int_0^1\frac{1}{1+x+x^2} \, dx <\frac{\pi}{4}$ Show that $$\frac{1}{3}<\int_0^1\frac{1}{1+x+x^2}\,dx <\frac{\pi}{4}$$
I want to use if $f<g<h$ then $\int f<\int g<\int h$ formula for Riemann integration.
$1+x^2<1+x+x^2$ and it will give RHS as $$\frac{1}{1+x+x^2}<\frac{1}{1+x^2}$$
How to choose... | Alternative approach: for any $x\in(0,1)$ we have
$$ \frac{1}{1+x+x^2}=\frac{1-x}{1-x^3}=(1-x)\sum_{n\geq 0}x^{3n}=\sum_{n\geq 0}x^{3n}-x^{3n+1} \tag{1}$$
hence
$$ \mathcal{J}=\int_{0}^{1}\frac{dx}{1+x+x^2} = \sum_{n\geq 0}\frac{1}{(3n+1)(3n+2)} \tag{2}$$
and the series in the RHS of $(2)$ can be approximated through ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2699005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
What kind of matrix is this and why does this happen? So I was studying Markov chains and I came across this matrix \begin{align*}P=\left( \begin{array}{ccccc}
0 & \frac{1}{4} & \frac{3}{4} & 0 & 0\\
\frac{1}{4} & 0 & 0 & \frac{1}{4} & \frac{1}{2}\\
\frac{1}{2} & 0 & 0 & \frac{1}{4}& \frac{1}{4... | You can draw a graph for which your matrix is the matrix of probabilities of state change (some weights on the edges).
Paint states A, D and E as white, and B and C as black. Then any possible move changes the color.
Since P^n is the probablility matrix of moving from X to Y in exacly n steps (or sum of products of we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2699410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "24",
"answer_count": 5,
"answer_id": 4
} |
Finding an isomorphism $\mathfrak{sp}(2) \to \mathbb{R}^3$ I am studying Lie groups and Lie algebras and I am trying to find an isomorphism between the groups/algebras and $\mathbb{R}^n$.
For $Sp(2,\mathbb{R})$, using the standard skew-symmetric matrix, and the condition
$$\begin{pmatrix}a & c \\ b & d \end{pmatrix} \b... | I just noticed something really dumb. I forgot to use the constraint
$$\begin{pmatrix}0 & a + d \\ -a -d & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0\end{pmatrix}$$
Which gives me $a = -d$.
Therefore, the matrices for $\mathfrak{sp}(2)$ are those of the form
$$\begin{pmatrix}a & b \\ c & -a\end{pmatrix}$$
Correc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2703682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prob. 10, Sec. 3.3, in Bartle & Sherbert's INTRO TO REAL ANALYSIS, 4th ed: Is this sequence convergent? Here is Prob. 10, Sec. 3.3, in the book Introduction to Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition:
Establish the convergence or the divergence of the sequence $\left( y_n \right)$, where
... | Yes, you are all right. Because an increasing and bounded real number sequence is convergent, it's the axiom of Real number: A bounded set on $R$ has supremum.
For calculate the limit, consider function $$f(x)=\frac{1}{x+1}\ \ \text{on}\ \ [0,1]$$it's integrable, and we let $0<\frac{1}{n}<\frac{2}{n}<……<\frac{n-1}{n}<... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2704730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
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.