Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Show that the last two decimal digits of a perfect square must be one of the following pairs. Show that the last two decimal digits of a perfect square must be one of the following pairs:
$00, e1, e4, 25, o6, e9$ ($e$ stand for even digit, $o$ for odd).
Solution is given, but I don't understand it at all.
| So the last two digits of $n^2$ clearly depend on the last two digits of $n$. There are $100$ possibilities.
Now $(50+n)^2=2500+100n+n^2=100(25+n)+n^2$ and the multiple of $100$ shows that this part can't affect the last two digits, which are therefore the same as the last two digits of $n^2$. So numbers which differ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2534461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Solving for $x$ an $x\log(x)$ kind of equation Given the following:
$$\frac{x \log\left(\frac{4x}{d}\right)}{y_0} = 10$$
where $d$ and $y_0$ are constants, how can I solve for $x$?
| For the equation
$$\frac{x \log\left(\frac{4x}{d}\right)}{y_0} = 10$$
multiply both sides by $4 y_{0}/d$ to obtain
$$\frac{4 x}{d} \, \log\left(\frac{4 x}{d}\right) = \frac{40 y_{0}}{d}.$$
Now using the Lambert W-function defined by $x e^{x} = W(x)$ with the property $e^{W(x)} = \frac{x}{W(x)}$ then
\begin{align}
\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2536713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Miscellaneous Olympiad Problem Two brothers sold a herd of sheep which they owned. for each sheep, they sold they received as many rubles as the number of sheep originally in the herd. The money was then divided in the following manner. First, the older brother took ten rubles followed by the younger brother taking ten... | If $N$ represents the total number of sheep that were originally in the flock, then the brothers were paid a total of $N^2$ rubles. From the description of how the proceeds were divided, we know that if $N^2$ is divided by 20, the remainder will be between 10 and 19 (inclusive). That is, there are nonnegative integer... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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prove factored polynomial has no real roots the polynomial $6x^3-18x^2-6x-6$ can be factored as $6(x-r)(x^2+ax+b)$ for some $a,b \in \Bbb{R}$ and where $r$ is a real root fo the polynomial. How would you prove that the polynomial $x^2+ax+b$ has no real roots. I know that you can do polynomial long division to get concr... | A non-calculus approach is possible, although not necessarily the most efficient.
Consider $$f(x) = x^3 - 6x^2 - x - 1.$$ Let $r$ be a real root of $f$, which we know must exist. Then long division yields $$\frac{f(x)}{x-r} = x^2 + (r-6)x + (r^2 - 6r - 1) + \frac{r^3 - 6r^2 - r - 1}{x-r},$$ and since $r$ is a root, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2542382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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I need help with geometry Let $ABCD$ be a convex quadrilateral. Let $|\measuredangle ABC|$=$|\measuredangle ACD|$ and $|\measuredangle ACB|$=$|\measuredangle ADC|$. Let $O$ be the circumcenter of triangle $BCD$, distinct from $A$. Prove, that $|\measuredangle OAC|$ is a right angle.
I'm supposed to use spiral similarit... | Particularly in the absence of better ideas, I like coordinates. Without loss of generality (as the setup is invariant under similarity transformations), you can use $A$ as the origin and $\overrightarrow{AB}$ as first unit vector. So you get
$$A=\begin{pmatrix}0\\0\end{pmatrix}\qquad
B=\begin{pmatrix}1\\0\end{pmatrix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2543190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $7^{100} - 3^{100}$ is divisible by $1000$
Prove that $7^{100} - 3^{100}$ is divisible by $1000$
Equivalently, we want to show that $$7^{100} = 3^{100} \pmod {1000}$$
I used WolframAlpha (not sure if that's the right way though) and found that $\varphi (250) = 100$.
So by Euler's theorem: $$7^{100} \equiv ... | \begin{eqnarray}
7^{100}-3^{100} &=& (10-3)^{100}-3^{100}\\
&=& \underbrace{{100\choose 0}10^{100}-{100\choose 1}10^{99}\cdot 3+...-{100\choose 97}10^3 \cdot 3^{97}}_{10^3k}+{100\choose 98}10^2 \cdot 3^{98} -{100\choose 99}10 \cdot 3^{99}+3^{100}-3^{100}\\
&=&1000k +50\cdot 99\cdot10^2 \cdot 3^{98} -100\cdot 10 \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2547583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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solution using synthetic geometry I managed to solve this problem only using complex numbers but I'd like to solve it using synthetic geometry and I can't.
Can someone help me to solve this problem using synthetic geometry?
Let $ABC$ an acute triangle with $AB > AC$ . Let $O$ its circumcenter and let $D$ the midpoint... | Hints:
Construct squares $EDPR$ and $FDQS$ outwardly on the sides $ED$ and $FD$ of $\triangle DEF$. Let the line $PQ$ intersect lines $AB, AC$ at $U, V$ respectively .
First show that $MD \perp \operatorname{line} PQ = \operatorname{line}\ UV$. Therefore if $l$ is the line through $A$ that is parallel to $MD$ then $l \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2550308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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I want to know the limits and convergence/divergence of $a_{n+1} = (a_n+3)/(a_n+4)$ I`ve the following iteratively defined sequence:
$a_1=1$ and for $n\ge 1$,
$$a_{n+1} = \frac{a_{n}+3}{a_n+4}$$
I want to know the limits and if it is convergent or divergent, but I don't know how to handle two $a_n$'s in the formula.
If... | After testing a couple $a_n$ by hand it appears to converge. So let's try to solve:
$$x = \frac{x+3}{x+4}$$
$$x(x + 4) = x+3$$
$$x^2 +3x - 3 = 0$$
$$x = \frac{1}{2}(\sqrt{21} - 3)$$
But if we want to be rigorous and prove its convergence:
$$a_n = \frac{p_n}{q_n}$$
$$\frac{p_{n+1}}{q_{n+1}} = \frac{\frac{p_n}{q_n}+3}{\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2551531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find matrix exponential $e^A$ I have the matrix $$A =\begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix}$$
and I have to find $e^A$
I've found two complex-conjugate eigenvalues $\lambda_{1,2} = \pm i$
so substracting $\lambda_1 = i$ from the matrix's diagonal I got:
$$A_1 = \begin{pmatrix} -i & 1 \\-1 & i \end{pmatri... | Answer
Let a,b $\in \mathbb{R}$ and
$$
\begin{align}
I &=
\begin{pmatrix}
1 & 0 \\
0 & 1 \\
\end{pmatrix} \\
J &=
\begin{pmatrix}
0 & 1 \\
-1 & 0 \\
\end{pmatrix}
\end{align}
$$
Then the following formula is true,
$$ \exp(aI + bJ) = \exp(a)[I \cos(b) + J \sin(b)]$$
In the special c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2552613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Evaluate $\int_0^a \sqrt{\frac{x^3}{a^3-x^3}} dx$. Evaluate following in terms of Gamma function:
$$\int_0^a \sqrt{\frac{x^3}{a^3-x^3}} dx$$
I don't know how to proceed. So, please tell the intuition behind the solution.
| Set $$x =a\sin^{\frac{2}{3}}(t) \implies dx = \frac{2a}{3}\frac{\cos t}{\sqrt[3]{\sin t}} dt$$ giving us: $$I = \int_{0}^{a}\sqrt{\frac{x^3}{a^3-x^3}} dx = \frac{2a}{3}\int_{0}^{\frac{\pi}{2}}\sqrt{\frac{a^3\sin^2t}{a^3\cos^2t}}\frac{\cos t}{\sqrt[3]{\sin t}} dt$$ $$=\frac{2a}{3}\int_{0}^{\frac{\pi}{2}}\sin^{\frac{2}{3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2553744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove $5^n + 2 \cdot 3^n - 3$ is divisible by 8 $\forall n\in \mathbb{N}$ (using induction)
Prove $5^n + 2 \cdot 3^n - 3$ is divisible by 8 $\forall n\in \mathbb{N}$
Base case $n = 1\to 5 + 6 - 3 = 8 \to 8 \mid 8 $
Assume that for some $n \in \mathbb{N}\to 8 \mid 5^n + 2 \cdot 3^n - 3$
Showing $8 \mid 5^{n+1} + 2 ... | Hint: Try reducing your expression mod 8. For example, what is $5^n$ mod 8? Since $5 \equiv 5$ and $5^2 = 25 \equiv 1$,
$5^n \equiv \begin{cases} 5, n \text{ odd} \\ 1, n \text{ even} \end{cases}$.
Now do the same for $3^n$, and add.
(This solution doesn't use induction.)
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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Find the relation of the sides. How can I find the following relation $\frac{AO}{r}$?
$BO = OC, AB = AC, AD = BD$
I tried to do the following $r = \frac{AO + OC + \sqrt{{OC}^{2}+{AO}^{2}}}{2} $ but I was not able to infer the relation.
$CD$ is touched by the inner circle.
| Let's define, as in the diagram below: $a=AG$, $b=GC$, $c=CH$. From Pythagoras' theorem applied to triangle $AOC$ we have:
$(a+r)^2+(b+r)^2=(a+b)^2$, that is:
$$
\tag{1}
ar+br+r^2=ab.
$$
As $AD=BD$, we also have $AH=BL=BJ=2OC-JC$, that is: $a+b+c=2b+2r-c$, or: $2c=2r+b-a$. On the other hand, triangles $EGC$ and $FCH$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2557034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Show that $\int_{0}^{\pi}{\sin^5(x)\over 1+\cos^3(x)}\mathrm dx =\ln{3}?$
$$\int_{0}^{\pi}{\sin^5(x)\over 1+\cos^3(x)}\mathrm dx =\ln{3}\tag1$$
How can we show that $(1)=\ln{3}$
$u=\sin^3{x}$ then $du=3\sin^2{x}\cos{x}dx$
$${1\over 3}\int{u\mathrm du\over \cos{x}+\cos^4{x}}\tag2$$
This is not a good substitution
| Without using integration by substitution or by parts:
Trigonometric algebra
We use the identity that $(1-\cos x)(1+ \cos^3x)=(1-\cos^2x)(1-\cos x+\cos^2x)$ so $$\frac{\sin^2x}{1+\cos^3x}=\frac{1-\cos x}{1-\cos x + \cos^2x} \implies \frac{\sin^5x}{1+\cos^3x}=\frac{\sin^3x(1-\cos x)}{1-\cos x + \cos^2x}$$ Now RHS is equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2559009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 3
} |
How to Simplify $ \frac{\sin(3x)+\sin^3(x)}{\cos(3x)-\cos^3(x)} $
Simplify $$ \frac{\sin(3x)+\sin^3(x)}{\cos(3x)-\cos^3(x)}.$$
The solution is : $-\cot(x)$
I tried to: $$\frac{\sin(2x)\cos(x)+\cos(2x)\sin(x)+\sin^3(x)}{\cos(2x)\cos(x)+\sin(2x)\sin(x)-\cos^3(x)}.$$
| Note that
$$\begin{align}\cos(3x)+i\sin(3x)&=(\cos(x)+i\sin(x))^3
\\&=\cos^3(x)+3i\cos^2(x)\sin(x)
-3\cos(x)\sin^2(x)-i\sin^3(x).
\end{align}$$
Therefore, after separating real and complex parts, we obtain
$$
\cos(3x)=\cos^3(x)
-3\cos(x)\sin^2(x),\quad
\sin(3x)=3\cos^2(x)\sin(x)
-\sin^3(x).
$$
Finally
$$\frac{\sin(3x)+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Distance from center of a spiral Consider the following curve:
*
*We start at the center and take a step of 1 unit length
*Then turn to the left by $\pi/3$ and take a step of $9/10$ units in length
*Then turn to the left by $\pi/3$ and take a step of $81/100$ units in
length
*Then turn to the left by $\pi/3$ and ... | Short answer: take advantage of the periodicity: steps taken three turns apart are parallel.
With more detail: each step you take corresponds to adding a term of the form
\begin{equation}
\left(\frac{9}{10}\right)^ke^{ik\frac{\pi}{3}},
\end{equation}
starting from $k=0$. The terminal point of our spiral is then
\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2567890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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let $\forall x \neq 0 \ \ \ f(x+\frac{1}{x})=x^2-\frac{1}{x^2}$ then find the $f(x)$
let $\forall x \neq 0 \ \ \ f(x+\frac{1}{x})=x^2-\frac{1}{x^2}$ then find the $f(x)$
My try :
$$x^2-\frac{1}{x^2}=(x+\frac{1}{x})(x-\frac{1}{x})$$
And $$(x-\frac{1}{x})^2 =(x+\frac{1}{x})^2-4$$
so we have :
$$f(t)=\pm t \sqrt{t^2-4... | Such function cannot exist. Take $x=2$, then we have $$f(2+1/2)= 4-1/4= 15/4$$ Now take $x=1/2$, then we have $$f(2+ 1/2)=1/4-4=-15/4$$But $-15/4\neq 15/4$. Hence such function cannot exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2569695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How can i solve $5^{2x-\frac{1}{3}x^2} < 5^{2-2x} * (5^\frac{1}{3})^{x^2}+24$? How can i solve it?
$$5^{2x-\frac{1}{3}x^2} < 5^{2-2x} * (5^\frac{1}{3})^{x^2}+24$$
I don't have idea how to solve it..
| We have,
$$5^{2x-\frac{1}{3}x^2} < 5^{2-2x} * (5^\frac{1}{3})^{x^2}+24$$
$$5^{2x-\frac{1}{3}x^2} - 5^{2-2x} \cdot (5^\frac{1}{3})^{x^2}- 24<0 $$
$$5^{2x-\frac{1}{3}x^2} - 5^{2-2x + \frac{1}{3} x^2 } - 24 < 0 $$
Substitute $ 5^{2x - \frac{1}{3}x^2} = t $,
$$t - \frac{5^2}{t} - 24 < 0 $$
$$\frac{(t-25)(t+1)}{t} <... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2570606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Wrong Wolfram Alpha result for $\lim_{(x,y)\to(0,0)}\frac{xy^4}{x^4+x^2+y^4}$?
I'm trying to solve this limit:
$$
\lim_{(x,y)\to(0,0)}\frac{xy^4}{x^4+x^2+y^4}
$$
Here's my attempt:
$$0 \le |\frac{xy^4}{x^4+x^2+y^4} - 0| = \frac{|x|y^4}{x^4+x^2+y^4},$$ and since $x^4+x^2 \ge0$ then $\frac{y^4}{x^4+x^2+y^4} \le 1$ s... | For
\begin{align*}
\lim_{(x,y)\rightarrow(0,0)}\dfrac{xy^{4}}{x^{4}+x^{2}+y^{2}},
\end{align*}
one does the following step which is similar to your technique:
\begin{align*}
\left|\dfrac{xy^{4}}{x^{4}+x^{2}+y^{2}}\right|\leq|xy^{2}|\cdot\dfrac{y^{2}}{x^{4}+x^{2}+y^{2}}\leq|x|\cdot y^{2}\rightarrow 0
\end{align*}
as $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Evaluate $\int_0^\infty \frac {(\log x)^4dx}{(1+x)(1+x^2)}$ Evaluate $$\displaystyle\int_0^\infty \frac {(\log x)^4dx}{(1+x)(1+x^2)}$$
This is a past final term exam problem of a complex analysis course at my university. I am studying for this year’s exam and I found this problem. The examiner assumes us to use residue... | For $-1\lt a\lt0$,
$$
\begin{align}
&\int_0^\infty\frac{x^a}{(1+x)(1+x^2)}\,\mathrm{d}x\\
&=\int_0^\infty\frac{x^a}2\left(\frac1{1+x}+\frac{1-x}{1+x^2}\right)\mathrm{d}x\\
&=\frac12\int_0^\infty\frac{x^a}{1+x}\,\mathrm{d}x+\frac14\int_0^\infty\frac{x^{\frac{a-1}2}}{1+x}\,\mathrm{d}x-\frac14\int_0^\infty\frac{x^{\frac{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2571670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Why is function domain of fractions inside radicals not defined for lower values than those found by searching for domain of denominator in fraction? Consider function $y = \sqrt{\frac{1-2x}{2x+3}}$. To find the domain of this function we first find the domain of denominator in fraction:
$2x+3 \neq 0$
$2x \neq -3$
$x \... |
Find the domain of $y=\sqrt{\frac{1-2x}{2x+3}}$
Draw a simple table:
$$
\begin{array}{c|ccccc}
x & \text{under $-3/2$} & \text{$-3/2$} & \text{$]-3/2;1/2[$} & \text{$1/2$} & \text{over $1/2$} \\
\hline
1-2x & + & + & + & 0 & - \\
2x+3 & - & 0 & + & + & + \\
\frac{1-2x}{2x+3} & - & /// & + & 0 & - \\
\sqrt{\frac{1-2x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2572000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 3
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Minimum distance from $2x^2 + 2xy + 4yz + z^2 = 1 $ to the origin Let Σ be the surface in $R^3$ given by
$2x^2 + 2xy + 4yz + z^2 = 1 $
By writing this equation as
$x^TAx$ = 1,
with A a real symmetric matrix, show that there is an orthonormal basis such that, if we use coordinates
$(u, v, w)$ with respect to this new ba... | Apply Lagrange Multipliers Method to minimize $f(x,y,z)= x^2+y^2+z^2$ subject to $g(x,y,z)= 2x^2+2xy+4yz+z^2-1$, results in $x=y=z=\frac13$ or $x=y=z=-\frac13$ with the minimum value of $f(x,y,z)=\frac13$. Therefor the minimum distance is $\sqrt{\frac13}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2573277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Relatively prime factors of $24500$
Let $N=24500$, then find the number of ways by which $N$ can be resolved into two coprime factors?
My tries:
$N=24500=2^2\cdot 5^3\cdot 7^2$, for co prime no those two factors of $24500$ should share something in common, so the two factors should be such that either one of the two ... | For integer $a,b,c>0$
$$2^a\cdot5^b\cdot7^c=(1\cdot2^a)(1\cdot5^b)(1\cdot7^c)$$
$=((1\cdot2^a5^b)$ or $(2^a\cdot5^b))(1\cdot7^c)$
$=(1\cdot2^a5^b7^c)$ or $(7^c\cdot2^a5^b)$ or $(2^a7^c,5^b)$ or $(2^a,5^b7^c)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2574089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Weird result proving : $1+2+3+...+n<\frac{(2n+1)²}{8} $
$$1+2+3+...+n<\frac{(2n+1)²}{8} $$
So, I'm going through Apostol's Calculus, and solving this:
I want to prove the aforementioned inequality, and proceed like this :
*
*$A(1) = 1 < \frac{9}{8}$
$A(2) = 1+2=3 < \frac{25}{8}$
$...$
*
*$A(k)=1+2+3+...+k<\frac{... | You are almost done. Note that you should show that
$$1+2+3+\dots+k+(k+1)<\frac{(2k+1)^2}{8}+(k+1)\color{red}{\leq }\frac{(2(k+1)+1)^2}{8}.$$
which implies that
$$1+2+3+\dots+k+(k+1)< \frac{(2(k+1)+1)^2}{8}.$$
So showing that
$$\frac{(2k+1)^2}{8}+(k+1)=\frac{(2(k+1)+1)^2}{8}$$
concludes the proof.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $x^3 +y^3 + z^3 =57$ How can we solve $x^3 + y^3 + z^3 =57$ efficiently in a shorter way. $x$ $y$ and $z$ are integers. Given that modulus of $x$ $y$ and $z$ is less than or equal to five. We can of course do by hit and trial but what is the method of solving such questions. I actually stumbled upon this equation... | With some forethought, you can bring it down to one case to check directly, which turns out to be a solution. A previous version of this answer examined the equation mod $7$, which led to $16$ cases to directly examine. This answer examines mod $9$, which works out even better. (The reason $7$ and $9$ are good moduli t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2576947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Property of Medians and Cirumcircle Let $ABC$ be a non-isosceles triangle. Medians of $\triangle ABC$ intersect the circumcircle in points $L,M,N$. If $L$ lies on the median of $BC$ and $LM=LN$, then prove that $2a^2=b^2+c^2$.
My Attempt:
Let $G$ be the centroid of $\triangle ABC$ and $D$ be the mid-point of $BC$.
Sin... |
Consider $\triangle AGB$ and $\triangle MGL$
$$\angle BGA=\angle LGM$$
$$\angle GAB=\angle GML$$
$$\triangle AGB \sim \triangle MGL$$
$$\frac{GB}{GL}=\frac{AG}{GM}=\frac{AB}{LM}$$
$$AG=\frac{c}{LM}GM$$
Similarly $\triangle AGC \sim \triangle NGL$
$$\frac{AG}{GN}=\frac{GC}{LG}=\frac{AC}{LN}$$
$$AG=\frac{b}{LN}GN$$
Thu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Two fair coins are tossed until both turn up heads A penny and a dime are tossed together until both turn up heads, after which no more tosses are made. Find the expected number of times the penny comes up heads.
What I've tried:
Let $X$ and $Y$ be the number of times the penny and dime come up heads, respectively. The... | Yes, there is an easier way. Suppose the expected number of times the penny comes up heads is $x$. Then consider the outcome of the first pair of tosses: if both come up heads, $x = 1$, with probability $1/4$.
With probability $1/2$, the penny comes up tails. Regardless of the outcome of the dime, we don't stop; bu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Compute $e^{Ax}$ for the $2 \times 2$ matrix $A$ Consider the real $2 \times 2$ matrix $A$ $$A=\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$$
I want to compute $e^{Ax}$.
I can see that $A^2=I$ which looks like it should be useful.
Using the definition of the matrix exponential $$e^{Ax}=I+Ax+\frac{(Ax)^2}{2!}+ \cdots$... | We have $$e^{Ax}=\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}\left(1+\frac{x^2}{2!}+ \cdots\right)+\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}\left(x+\frac{x^3}{3!}+ \cdots\right)$$ This can be written as $$\begin{pmatrix} 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+... & 0 \\ 0 & 1+(-x)+\frac{x^2}{2!}+\frac{(-x)^3}{3!}+...\end{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2577915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Check if the sequence converges using Cauchy criterion I need to check if the sequence of $x_n = \dfrac1{\sqrt1} + \dfrac1{\sqrt2} ... + \dfrac1{\sqrt n}$ converges using the Cauchy criterion.
Obviously, first I have to do is to somehow check if $|x_n - x_{n+p}| $ (where $p$ is any positive whole number) is convergent... | For any $n \in \mathbb{N}$ we have:
\begin{align}
x_{n^2} - x_n &= \left(\frac{1}{\sqrt{1}} +\ldots + \frac{1}{\sqrt{n^2}}
\right) - \left(\frac{1}{\sqrt{1}} +\ldots + \frac{1}{\sqrt{n}}
\right)\\
&= \frac{1}{\sqrt{n+1}} + \frac{1}{\sqrt{n+2}} + \ldots + \frac{1}{\sqrt{n^2}}\\
&\ge \underbrace{\frac{1}{\sqrt{n^2}} +... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Factoring $x^7+3x^6+9x^5+27x^4+81x^3+243x^2+729x+2187$
Question: How would you factor$$P(x)=x^7+3x^6+9x^5+27x^4+81x^3+243x^2+729x+2187$$
I thought for a while and realized that the coefficients are in powers of $3$, so $x=-3$ is a factor. Taking that factor out, we see that the septic is equal to$$P=(x+3)(x^2+9)(x^4+... | For the sake of an alternative, less clever approach: pretending to not notice the pattern of increasing powers of $3$, the root $x=-3$ can also be found by brute force using the rational root theorem. Quite obviously, the polynomial has no positive roots, so it's enough to try the negative divisors of $2187=3^7$, whic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2580330",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
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How to algebraically solve $\frac{1}{x} < 5$ inequality to obtain two solutions? Given the inequality:
$\frac{1}{x} < 5$
In order to find a solution, I would normally multiply both sides by $x$:
$1 < 5x$
Then I would divide by $5$
$\frac{1}{5} < x$
To obtain the solution: $x > \frac{1}{5}$.
Now, the thing is, the solut... | Note that by going from $\dfrac{1}{x} < 5$ to $1 < 5x$, you are assuming that $x > 0$. You see that by assuming $x > 0$, you obtain $x > \dfrac{1}{5}$.
Now assume that $x < 0$. Then $\dfrac{1}{x} < 5 \implies 1 > 5x\implies x < \dfrac{1}{5}$ (flipping the inequality). But, remember, we assumed that $x < 0$. Thus, if $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2581066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 5,
"answer_id": 1
} |
Find all real $x$ for which $\log_{10}(2^{x-1}+3^{x+1})=2x-\log_2{3^x}$ My attempt:
$$\log_{10}(2^{x-1}+3^{x+1})=x\cdot(2-\log_2{3})$$
I am stuck after this. If the terms inside LHS were a product, this problem would have been a piece of cake. The domain of $x$ is all real here so I can't limit the range of $x$ for a h... | Hint: power each side by 2.
$$A = 2^{x-1} + 3^{x+1} = 2^{2x-xlog_2(3)}$$
$$= (2^{2-log_2(3)})^x = \left(\frac{4}{3}\right)^x$$
$$\Rightarrow 2^{x-1}3^x+3^{2x+1} = 2^{2x}$$
Suppose $X = 2^x$, $Y=3^x$:
$$\frac{1}{2}XY+ 3Y^2 = X^2$$
$$ X^2 - 3Y^2 - \frac{1}{2}XY = 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2583185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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To find the sum of $1+\frac13-\frac12-\frac14-\frac16+\frac15+\frac{1}{7}-\frac{1}{8}-\frac{1}{10}-\frac{1}{12}+\ldots$ I have to find the sum of:
$$1+\frac13-\frac12-\frac14-\frac16+\frac15+\frac{1}{7}-\frac{1}{8}-\frac{1}{10}-\frac{1}{12}+\ldots$$
My attempt:
$$\left( 1+\frac13-\frac12-\frac14-\frac16\right)+\left(\f... | The series is not absolutely convergent; therefore, you cannot move around terms freely.In other words,
\begin{eqnarray*}\sum_{n=0}^{+\infty}\left(\frac{1}{4n+1}+\frac{1}{4n+3}-\frac{1}{6n+2}-\frac{1}{6n+4}-\frac{1}{6n+6}\right)\neq \sum_{n=0}^{+\infty}\left(\frac{1}{4n+1}+\frac{1}{4n+3}\Bigg)-\\-\sum_{n=0}^{\infty}\B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2585162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Count rational numbers in lowest terms in $[0,1]$ I was wondering if there is a formula for the count of rational numbers $\frac{p}{q}$ in lowest terms in $[0,1]$ similar to the sequence in this answer.
For example:
*
*For $q = 1$ we have two: $\{\frac{0}{1}, \frac{1}{1}\}$
*For $q = 2$ we have one: $\{\frac{1}{2}\... | Your question is related to Euler's totient function. In particular, the total number of fractions $p/q$, where $1\le p < q$, and $\gcd(p,q)=1$ equals $\varphi(q)$, where $\varphi(\cdot)$ is the Euler's totient function. Consequently, the required number of fractions is
$$2+\sum_{q=2}^{N} \varphi(q).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Is 2018 special because of these properties? I discovered that: $$2018=(6^2)^2+(5^2)^2+(3^2)^2+(2^2)^2$$
We also have: $$13^2+43^2=2018$$
And we have:
$$2018=44^2+9^2+1^2$$
I somehow tend to believe that there could be a finite number of these numbers that are sum of two squares, three squares and four fourth powers.
... | One infinite set is $2018k^4$ for any natural $k$. I strongly suspect that there are plenty more. Numbers that are a sum of three squares are very common. Numbers that are a sum of two squares are not so rare, so I would just start picking sums of four cubes and try to satisfy the other two. Another example is $$1^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2586660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
} |
Given $3\cos x - 4 \sin x = 2$, find $3 \sin x + 4 \cos x$ without first solving for $x$ If $$3\cos{x}-4\sin{x}=2$$
find $$3\sin{x} +4\cos{x} $$
I have solved the equation for $x$, then calculated the required value, but I think there is a direct solution without solving the equation.
| Hint:
Using Brahmagupta-Fibonacci Identity,
$$(a\cos x-b\sin x)^2+(a\sin x+b\cos x)^2=(a^2+b^2)(\cos^2x+\sin^2x)=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Inequality: $\frac{1+x}{1-x} > e^{2x}$ for $0 < x < 1$ I was asked to prove the inequality $\frac{1+x}{1-x} > e^{2x}$ for $0 < x < 1$
My approach would be with the differential equation.
$f(x) = e^{2x} - \frac{1+x}{1-x} $
$f(x)' = 2e^{2x} − \frac{x+1}{(1−x)^2} − \frac{1}{1−x} = 2e^{2x}−\frac{2}{(1−x)^2} $
Now it would ... | $$\frac{1+x}{1-x} > e^{2x}\iff \ln\left(\frac{1+x}{1-x}\right)>2x$$
$$\ln\left(\frac{1+x}{1-x}\right)=\ln(1+x)-\ln (1-x)>x-\frac{x^2}2-\left(-x-\frac{x^2}2 \right)=2x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2588990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 0
} |
Determine exponential generating function of the case given I have a problem to determine EGF that show how many ways we can distribute r different people into n different rooms such that every room has at least 2 people and no more than 5 people. Of course, we know that this is a permutation case (and that's why it le... |
Using the coefficient of operator $[x^j]$ to denote the coefficient of $x^j$ in a series the wanted number is
\begin{align*}
r![x^r]&\left(\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}\right)^n\\
&=r![x^r]\frac{x^{2n}}{5!}\left(60+20x+5x^2+x^3\right)^n\\
&=\frac{r!}{5!}[x^{r-2n}]\left(60+20x+5x^2+x^3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2589731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to obtain the following definite integral? I have no idea how to obtain the answer to the following problem:
$$I=\int_{0}^{1}\frac{dx}{\sqrt{8-x^2-x^3}}$$
The answer has been given as
$$\sin^{-1} \frac{1}{2\sqrt{2}}< I < \frac{1}{\sqrt{2}}\sin^{-1} \frac{1}{2}$$
Any help will be appreciated. Thanks in advance.
| $I$ is an elliptic integral, accurate approximations can be deduced from convexity. Over $(0,1)$ we clearly have $8-x^2-x^3 > 8-2x^2$, hence
$$ I < \int_{0}^{1}\frac{dx}{\sqrt{8-2x^2}} = \frac{\pi}{6\sqrt{2}} \tag{Ub}$$
but we also have $\frac{1}{\sqrt{8-x^2-x^3}}>\frac{1}{2\sqrt{2}}\left(1+\frac{x^2+x^3}{16}\right)$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2591926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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Solve $y^{(4)}-2y^{(3)}+2y'-y=xe^x.$ Solve $y^{(4)}-2y^{(3)}+2y'-y=xe^x.$
The characteristic equation is $(r-1)^3(r+1)\Rightarrow y_h=(C_1+C_2x+C_3x^2)e^x+C_4e^{-x}.$
The problem is the particular equation. Why doesn't it work with the ansatz
$$y=(ax+b)e^x?$$
I get
\begin{array}{lcl}
y & = & e^x(ax+b) \\
y' ... | Solve step by step the equation $$(D-1)^3(D+1)y=xe^x$$ (where $D$ is differentiation operator) and there is no need to worry about guessing particular solution. Let $z=(D-1)^3y$ so that $$(D+1)z=xe^x$$ or $$D(e^xz) =xe^{2x}$$ or $$ze^x=\frac{xe^{2x}}{2}-\frac{e^{2x}}{4}+a$$ or $$z=\frac{xe^x} {2}-\frac{e^x}{4}+ae^{-x}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2594875",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 0
} |
Evaluate a limit at infinity Find the following limit without Lospital rule nor series
$$\lim_{n\to \infty}\left(\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\right)^{\frac{1}{n}}$$
I tried to take log to both sides to be
$$\ln L=\lim_{n\to \infty}\frac{\ln\left((ab)^n +(bc)^n+(ac)^n\right)}{n}-\lim_{x\to \infty}\frac{n\l... | Assume $a,b,c>0$ and wlog $\frac1a=max\{\frac1a,\frac1b\frac1c\}$ thus
$$\left(\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\right)^{\frac{1}{n}}=e^{\frac{\log{\left(\frac{1}{a^n}+\frac{1}{b^n}+\frac{1}{c^n}\right)}}{n}}=e^{\frac{\log{\left(\frac{1}{a^n}\right)}+\log{\left(1+\frac{a^n}{b^n}+\frac{a^n}{c^n}\right)}}{n}}\to\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2598285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Find $4762^{5367} \pmod{13}$
Find $4762^{5367} \pmod{13}$
So I started off and got:
$$4762 = 4 \pmod{13}$$
Therefore, $$4762^{5367} = 4^{5367} \pmod{13}$$
But I did not know how to proceed after this.
Any help?
| For a small modulus like $13$, it's worth just looking for the pattern that $4$ follows on exponentiation:
$\bmod 13: \\
\begin{align}
\qquad
4^0 &\equiv 1\\
4^1 &\equiv 4\\
4^2 &\equiv 16 \equiv 3\\
4^3 &\equiv 12 \equiv -1\\
4^4 &\equiv -4 \equiv 9 \\
4^5 &\equiv -3 \equiv 10\\
4^6 &\equiv -1^2 \equiv 1\\
\end{align... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2599020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 2
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Which graph corresponds to which equation? The figure below represents the graphs of parametric curve $x(t) = \sin(t)$, $y(t) = \frac{1}{2}\sin(2t)$, $0 \le t \le 2\pi$ and polar curve $r^2 = \cos(2\theta)$, $0\le \theta \le 2\pi$. Which curve corresponds to which equation?
I simply have no idea how to approach this, a... | The point $(x,y)=(1,0)$ belongs to both curves, at $t=\pi/2$ for the parametric curve and at $\theta=0$ for the polar curve, so let us compare the behaviour of the two curves in a neighborhood of this point.
Our only prerequisite shall be the one term Taylor expansion $$\sqrt{1-2z}=1-z+o(z^2)$$ when $z\to0$, which (if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2601374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to prove that this triangle is equilateral? Question: If $\cos A +\cos B +\cos C=\frac{3}{2}$, prove that the triangle is equilateral.
My attempt: According to cosine rule, $\cos A=\frac{b^2+c^2-a^2}{2bc}$, $\cos B=\frac{c^2+a^2-b^2}{2ca}$ and $\cos C=\frac{a^2+b^2-c^2}{2ab}$.
$\cos A +\cos B +\cos C=\frac{3}{2}$
$... | A slick way is to combine Carnot's theorem with Euler's inequality $R\geq 2r$.
$$ R\cos A+R\cos B + R\cos C $$
is the sum of the signed distances of the circumcenter from the triangle sides, and it equals $R+r$. Since $r<\frac{R}{2}$ unless $ABC$ is equilateral,
$$ R\cos A+R\cos B + R\cos C = \frac{3}{2}R$$
implies tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2602051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
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... | It remains to show that $7^{4n} - 1$ is a multiple of $600$.
Since $600 = (2^3)(3)(5^2)$, the goal is equivalent to showing that the three congruences
\begin{align*}
7^{4n} &\equiv 1\;(\text{mod}\;2^3)\\[4pt]
7^{4n} &\equiv 1\;(\text{mod}\;3)\\[4pt]
7^{4n} &\equiv 1\;(\text{mod}\;5^2)\\[4pt]
\end{align*}
hold for all ... | {
"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": 1
} |
Range of $S,T$ in $2$ sums
If $\displaystyle S = \sum^{4n-1}_{k=3n}\bigg(\frac{k^2-7kn+13n^2}{n^3}\bigg)$ and $\displaystyle T= \sum^{4n}_{k=3n+1}\bigg(\frac{k^2-7kn+13n^2}{n^3}\bigg)$. then which one is/are
right $\; (a)\displaystyle \; S<\frac{5}{6}\; (b)\; T<\frac{5}{6}\; (c)\; S>\frac{5}{6}\; (d)\; T>\frac{5}{6}$... | First,\begin{align*}
S - T &= \sum_{k = 3n}^{4n - 1} \frac{1}{n^3} (k^2 - 7kn + 13n^2) - \sum_{k = 3n + 1}^{4n} \frac{1}{n^3} (k^2 - 7kn + 13n^2)\\
&= \frac{1}{n^3} ((3n)^2 - 7 \cdot 3n \cdot n + 13n^2) - \frac{1}{n^3} ((4n)^2 - 7 \cdot 4n \cdot n + 13n^2)\\
&= \frac{1}{n} - \frac{1}{n} = 0.
\end{align*}
Now,\begin{ali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that if $x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1$ then $xyz=0$ For any real numbers $x,y,z$ such that $x+y+z = x^2+y^2+z^2 = x^3+y^3+z^3 =1\\$ show that $x \cdot y \cdot z=0$.
I think that because $x \cdot y \cdot z = 0$ at least one of the three number should be equal to zero, but I'm stuck into relating the other things... | Consider $f(w)=(w-x)(w-y)(w-z).$ We have $f(w)=w^3-Aw^2+Bw-C$ where $A=x+y+z$ and $B=xy+yz+zx$ and $C=xyz.$
Suppose $x+y+z=x^2+y^2+z^2=x^3+y^3+z^3=1.$ We have $0=f(x)=f(y)=f(z).$ Hence $$0=f(x)+f(y)+f(z)=$$ $$=(x^3+y^3+z^3)-A(x^2+y^2+z^2)+B(x+y+z)-3C=$$ $$=1-A+B-3C=$$ $$=1-(x+y+z)+(xy+yz+zx)-3C=$$ $$=1-1+(xy+yz+zx)-3C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2612380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 8,
"answer_id": 5
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Span of Integer Linear Combinations This question relates to linear combinations of vectors; however, vectors can only be scaled by integer values. This means that if we have two vectors $\binom{a}{b}$ and $\binom{c}{d}$, we can only have $$x\binom{a}{b} + y\binom{c}{d},$$ where $a, b, c, d, x,y\in \mathbb{Z}.$
I was ... | Claim: $\begin{pmatrix}a\\b\end{pmatrix}$ and $\begin{pmatrix}c\\d\end{pmatrix}$ span the whole $\mathbb Z^2=M_{12}(\mathbb Z)$ if and only if $\det\left[\begin{matrix}a&c\\b&d\end{matrix}\right]=ad-bc=\pm 1$.
Proof sketch: basically, the span is the whole of $\mathbb Z^2$ if and only if $\begin{pmatrix}1\\0\end{pmatri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2613869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determinant equal to zero : elements are angles Let $\alpha, \beta, \gamma$ be internal angles of an arbitrary triangle.
I want to show that $$\det \begin{pmatrix}\cos \beta & \cos \alpha&-1 \\ \cos\gamma & -1 & \cos\alpha \\ -1 & \cos\gamma & \cos\beta\end{pmatrix}=0$$
We have the following:
\begin{align*}&\det \be... | the product of the three cosines is given by $$-\frac{\left(a^2-b^2-c^2\right)
\left(a^2+b^2-c^2\right) \left(a^2-b^2+c^2\right)}{2
a^2 b^2 c^2}$$
for your second sum we get
$$2 \left(\frac{\left(a^2+b^2-c^2\right)^2}{4 a^2
b^2}+\frac{\left(a^2-b^2+c^2\right)^2}{4 b^2
c^2}+\frac{\left(-a^2+b^2+c^2\right)^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2614980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to find major & minor axes of the ellipse $10x^2+14xy+10y^2-7=0$?
What are major & minor axes of the ellipse: $10x^2+14xy+10y^2-7=0$ ?
My trial:
from given equation: $10x^2+14xy+10y^2-7=0$
$$10x^2+14xy+10y^2=7$$
$$\frac{x^2}{7/10}+\frac{xy}{7/14}+\frac{y^2}{7/10}=1$$
I know the standard form of ellipse: $\frac{... | Yet another method is with matrices and linear algebra. If we stuff $(x,y,1)^T$ into a vector, then the matrix
$${\bf M}=\left[\begin{array}{ccc}10&7&0\\7&10&0\\0&0&-7\end{array}\right]$$
Can be used to express the ellipse as a scalar product with matrix multiplication:
$$\left[\begin{array}{ccc}x&y&1\end{array}\right... | {
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"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
How can I calculate $\lim\limits_{x\rightarrow 0} \frac{(1+x)^{\frac{1}{x}}-e}{x}$?
How can I calculate this limit?
$$\lim_{x\rightarrow 0} \frac{(1+x)^{\frac{1}{x}}-e}{x}$$
I thought about L'Hospital because case of $\frac{0}{0}$, but I don't know how to contiune from this point..
| By inequality
$$x-\frac{x^2}{2}\leq \log (1+x) \leq x-\frac{x^2}{2}+\frac{x^3}{3}$$
$$1-\frac{x}{2}\leq \frac{\log (1+x)}{x} \leq 1-\frac{x}{2}+\frac{x^2}{3}$$
we have that
$$e^{1-\frac{x}{2}}\leq (1+x)^\frac{1}{x}=e^{\frac{\log (1+x)}{x}} \leq e^{1-\frac{x}{2}+\frac{x^2}{3}} $$
thus
$$-\frac{e}2\le\frac{e^{1-\frac{x}{... | {
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"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 5
} |
Proof verification: $\forall x \in [\frac{\pi}{2}, \pi], sin(x) - cos(x) \geq 1$. Would someone be willing to verify the following proof by contradiction?
Theorem: $\forall x \in [\frac{\pi}{2}, \pi], sin(x) - cos(x) \geq 1$.
Suppose, for the sake of contradiction, that this statement is not true. Then, $\exists x \in... | Use that
$$\cos x=\sin \left( \frac{\pi}{2} -x\right)$$
and the sum-to-product formulas.
Notably from
$$\sin \theta - \sin \varphi = 2 \sin\left( \frac{\theta - \varphi}{2} \right) \cos\left( \frac{\theta + \varphi}{2} \right)$$
we obtain
$$\sin x - cos x=\sin x -\sin \left( \frac{\pi}{2} -x\right)=2\sin\left(x-\frac{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Evaluating: $\int \frac {1+\sin (x)}{1+\cos (x)} dx$
Evaluate: $\int \dfrac {1+\sin (x)}{1+\cos (x)} dx$
My Attempt:
$$=\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 (\... | You were almost there, just substitute $t=\tan(\frac x2)$ now.
$\displaystyle\dfrac 12\int \left(1+\tan(\dfrac x2)\right)^2\mathop{dx}=\int \dfrac{(1+t)^2}2\times\dfrac{2\mathop{dt}}{1+t^2}=\int \left(1+\dfrac{2t}{1+t^2}\right)\mathop{dt}=t+\ln(1+t^2)+C$
Remark that you can substitute directly without intermediate trig... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 0
} |
System of equations involving complex numbers I'm getting confused to figure this work out. The only thing that came into my head was using AM-GM inequality, but i just get stuck. Here's the problem:
Let $a,b,c,$ be the complex numbers such that $abc=1$. and
\begin{cases}
a^{20}+b^{20} + c^{20} &= \frac{1}{a^{20}} + \... | Actually,
Lemma 1: if $abc=1$, and if $a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, then one of $a,b,c$ is 1.
Proof: multiply by $abc=1$:
$$a+b+c=ab+bc+ac\Leftrightarrow abc-ab-ac-bc+a+b+c-1=0\Leftrightarrow (a-1)(b-1)(c-1)=0$$
Lemma 2: If $\gcd(u,v)=1$, and $a^u=a^v=1$, then $a=1$.
Proof: $1=mu+nv$ for some $m,n\in\mat... | {
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"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Computing the definite integral $\int _0^a \:x \sqrt{x^2+a^2} \,\mathrm d x$
Compute the following definite integral $$\int _0^a \:x \sqrt{x^2+a^2} \,\mathrm d x$$
This is what I did:
$u = x^2 + a^2 $
$du/dx = 2x$
$du = 2xdx$
$1/2 du = x dx$
$\int _0^a\:\frac{1}{2}\sqrt{u}du = \frac{1}{2}\cdot \frac{u^{\frac{3}{2}}}{... | Hint
You made mistakes here:
$$\frac{1}{3}\cdot \left(x^2+a^2\right)^{\frac{3}{2}}\big |_0^a=\frac 1 3 (2a^2)^{\frac 3 2}-\frac {a^3}{3}=\frac 1 3 (\sqrt{2^3}|a|^3)-\frac {a^3}{3}=....$$
You were almost done...
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
The number of ways in which one can choose three distinct numbers from the set so that the product of the chosen numbers is divisible by $9$
Consider the set $A=\{1,2,3,...,30\}$
. The number of ways in which one can choose three
distinct numbers from $A$ so that the product of the chosen numbers is divisible b... | You haven't adressed "three distinct numbers" correctly. For instance, you aren't allowed to pick $3, 3, 5$, so the first term shouldn't be $7\cdot 7\cdot 20$, it should be $7\cdot 6\cdot 20$. And then, still, you count $3, 6, 5$ as a separate choice from $6, 3, 5$. Therefore, what you really want is $\frac{7\cdot 6}2\... | {
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"url": "https://math.stackexchange.com/questions/2622034",
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"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Computing $\lim\limits_{x\to0}\int_0^x\frac{t^2}{(x-\sin x)\sqrt{a+t}}\,dt$ without L'Hopital I am computing the following limit
$$\lim_{x\to0}\int_0^x\frac{t^2}{(x-\sin x)\sqrt{a+t}}\,dt$$
where $a$ is a parameter. the case $a= 0$ has been resolved. But yet, for the case $ a\neq 0$ I want to compute it without using ... | With Maclaurin series:
$$\lim_{x\to0}\int_0^x\frac{t^2}{(x-\sin x)\sqrt{a+t}}\,dt=\lim_{x\to0}\frac{1}{(x-\sin x)}\int_0^x\frac{t^2}{\sqrt{a+t}}\,dt=\lim_{x\to0}\frac{1}{(x-\sin x)}\left[\frac{2}{15}\sqrt{a+t}(8a^2-4at+3t^2)\right]_{0}^{x}=\lim_{x\to0}\frac{1}{(x-\sin x)}\frac{2}{15}\left[\sqrt{a+x}(8a^2-4ax+3x^2)-\sqr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2623247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
The Arithmetic Mean (A.M) between two numbers exceeds their Geometric Mean (G.M.) The Arithmetic Mean (A.M) between two numbers exceeds their Geometric Mean (G.M.) by $2$ and the GM exceeds the Harmonic Mean (H.M) by $1.6$. Find the numbers.
My Attempt:
Let the numbers be $a$ and $b$. Then,
$$A.M=\dfrac {a+b}{2}$$
$$G.... | I don't know
why I do this,
but here is
the general case.
Suppose
$am = gm+u$
and
$gm = hm+v$
with
$u, v \ne 0$
and
$u \ne v$.
Since
$hm \le gm
\le am
$,
$u \ge 0$
and
$v \ge 0$.
Since
$gm^2 = am\cdot hm$,
$gm^2
=(gm+u)(gm-v)
=gm^2+gm(u-v)-uv
$
so
$gm(u-v) =uv$.
Therefore
$u > v$
and
$gm
=\dfrac{uv}{u-v}
$.
Then
$am
=g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2625855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Equation System with 4 real variables I need to solve the next equation system:
Find all real numbers $a,b,c,d$ such that:
$$
\left\{
\begin{array}{c}
a+b+c+d=20 \\
ab+ac+ad+bc+bd+cd=150 \\
\end{array}
\right.
$$
I tried something like this:
$b+c+d=20-a$
And i put the second equation like this
$a(b+c+d) + bc+bd+cd=... | $$0=3(a+b+c+d)^2-8(ab+ac+ad+bc+bd+cd)=$$
$$=(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^2\geq0,$$
where the equality occurs for $a=b=c=d.$
Thus, $a=b=c=d=5.$
| {
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"url": "https://math.stackexchange.com/questions/2627063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Minimum value of $(a+1)(b+1)$
If $a,b$ are the roots of the equation $$(\arcsin x+\arctan x)p^2+25p\pi+2(\arccos x+\operatorname{arccot} x)=0$$
Then minimum of $(a+1)(b+1)$
Try: $$a+b=\frac{25\pi}{\arcsin x+\arctan x}$$ and $$ab=\frac{2(\arccos x+\operatorname{arccot} x)}{\arcsin x+\arctan x}$$
So $$ab+a+b+1=\frac{25... | $x\neq0$, otherwise our equation has unique root.
Also, $$\left(25\pi\right)^2-8(\arcsin{x}+\operatorname{arctan}x)(\arccos{x}+\operatorname{arccot}x)\geq$$
$$\geq\left(25\pi\right)^2-8\left(\frac{\arcsin{x}+\operatorname{arctan}x+\arccos{x}+\operatorname{arccot}x}{2}\right)^2=625\pi^2-2\pi^2>0,$$
which says that our ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to get first derivative? My question is to find the first derivative using product and/or quotient rule
$$f(x) = {(x^2+1)(x^2-2) \over 3x+2}.$$
The solution to the problem is:
$$f'(x) = {[(2x)(x^2-2)+(x^2+1)(2x)](3x+2)-(3)[(x^2+1)(x^2-2)] \over (3x+2)^2}.$$
I'm having problems getting the same solution would someo... | When you have expressions which just contains products, quotients and powers, logarithmic differentiation makes life easier.
For your case
$$f = {(x^2+1)(x^2-2) \over 3x+2}\implies \log(f)=\log(x^2+1)+\log(x^2-2)-\log(3x+2)$$ Differentiate both sides
$$\frac{f'}f=\frac{2x}{x^2+1}+\frac{2x}{x^2-2}-\frac 3{3x+2}=\frac{9 ... | {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Value of $S_{10}+I_{10}$ Let $S_n=\sum_{i=1}^n \frac{1}{k}$ and$$I_n=\int_{1}^{n} \frac{x-[x]}{x^2} dx$$ then what is the value of $S_{10}+I_{10}$ ?
$S_{10}=1+1/2+1/3+.....+1/10$ and $$I_{10}=\int_{1}^{10} \frac{x-[x]}{x^2} dx$$
$$=ln 10-\int_{1}^{10} \frac{[x]}{x^2} dx$$
But I am stuck here.How to proceed f... | You just have to use
$$
\begin{align*}
\int_{1}^{10} \frac{[x]}{x^2} dx & =\int_1^2\frac{1}{x^2} dx+2\int_2^3\frac{1}{x^2} dx +3\int_3^4\frac{1}{x^2} dx+\cdots +9\int_9^{10}\frac{1}{x^2} dx \\
& =\sum_{k=1}^{9} k \int_k^{k+1}\frac{1}{x^2} dx \\
& = \sum_{k=1}^{9} k [-\tfrac{1}{u}]_k^{k+1} \\
&= \sum_{k=1}^{9} ... | {
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"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Find the volume of the following solid. I need to find the volume of that solid:
$$(x^2+y^2+z^2)^3=4y^2z^2$$
I obviously tried spherical coordinates, but that simply led me to nowhere. I used:
$$\begin{cases}x=r\cos\alpha \cos\beta\\[2ex]y=r\sin\alpha \cos\beta\\[2ex]z=r\sin\beta\end{cases}$$
and it got me to the point... | In spherical coordinates $(R,\theta,\phi)$ the Jacobian of the spherical coordinates is $R^2\sin\theta$. Also the equation of the surface enveloping the solid is:$$R^2=4\sin^2\theta\sin^2\phi\cos^2\phi=\sin^2\theta\sin^2(2\phi)$$
For calculating the volume we have
$$\iiint_VR^2\sin\theta dR d\theta d\phi=
\int_{0}^{\pi... | {
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"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
how can I calculate: $\lim_{n \to \infty} \frac{\frac{1}{n}-\ln(1+\frac{1}{n})}{n^{\frac{1}{n}}-1} $?
how can I calculate:
$$\lim_{n \to \infty} \frac{\frac{1}{n}-\ln(1+\frac{1}{n})}{n^{\frac{1}{n}}-1} $$
I tried with Hospital and it's not working. Can help please ?
| Using Taylor expansion
We have $$\frac{1}{n}-\ln\left(1+\frac{1}{n}\right)= \frac{1}{2n^2} +o\left(\frac{1}{n^2}\right)$$
since $$\ln(x+1) = x-\frac{x^2}{2} +o(x^2)$$
and similarly we have,
$$n^{1/n} = \exp(n\ln (1/n)) =\exp\left(n\left(\frac{1}{n}-\frac{1}{2n^2}+o\left(\frac{1}{n^2}\right)\right)\right) = 1 -\frac{1... | {
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"url": "https://math.stackexchange.com/questions/2630173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Formatting Functions To Avoid Loss Of Significant
Rewrite the following to avoid loss of significant
*
*$\ln(x+1)-\ln(x)$ where $x>>1$
*$\cos^2(x)-\sin^2(x)$ where $x\approx \frac{\pi}{4}$
*$\sqrt{x^2+1}-x$ where $x>>1$
*$\sqrt{\frac{1+\cos x}{2}}$
*
*Using taylor expansion we get $$x-\frac{x^2}{2}+\frac{x^... | You don't want to use the Taylor series when $x$ is large. Better to use the law of exponents to write $\log(x+1) - \log (x)=\log(1+\frac 1x)$ and use the Taylor series from there.
This is the usual approach. You want to analytically subtract the large parts of the two numbers, reducing the cancellation.
| {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find all $x \in \mathbb R$ such that $x + \sqrt{3}$ and $x^2 + \sqrt{3}$ are rational. Find all $x \in \mathbb R$ such that $x + \sqrt{3}$ and $x^2 + \sqrt{3}$ are rational. I have started by assuming that $x + \sqrt{3} = \frac{a}{b}$ and substituting $x = \frac{a}{b} - \sqrt{3}$ into $x^2 + \sqrt{3}$. It led me to $\f... | Let $x+\sqrt3=r$ and $x^2+\sqrt3=q$.
Thus, $$(r-\sqrt3)^2=q-\sqrt3$$ or
$$r^2-2\sqrt3r+3=q-\sqrt3,$$ which gives $2r=1$ and $r^2+3=q$ and $x=\frac{1}{2}-\sqrt3.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2631663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Point in triangle Inside a random triangle, consider a point P and its distances PM, PN and PQ to the 3 sides a, b and c.
For which location of the point P the below sum gets minimal?
$$\frac{a}{PM} + \frac{b}{PN} + \frac{c}{PQ}$$
I think it is the incenter but how do we prove it?
|
Note that
\begin{align}
a\cdot x+b\cdot y+c\cdot z&=2\,S=\mathrm{const}\quad\text{for fixed }a,b,c.
\end{align}
\begin{align}
\operatorname*{argmin}_{P}\left(\frac{a}x + \frac{b}y + \frac{c}z\right)
&=
\operatorname*{argmin}_{P}2\,S\cdot\left(\frac{a}x + \frac{b}y + \frac{c}z\right)
\\
&=
\operatorname*{argmin}_{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Find $g(2002)$ given $f(1)$ and two inequalities It is given that $f(x)$ is a function defined on $\mathbb{R}$, satisfying $f(1)=1$ and for any $x\in \mathbb{R}$,
$$f(x+5)\geq f(x)+5,$$
$$f(x+1)\leq f(x)+1.$$
If $g(x)=f(x)+1-x$ then find $g(2002)$.
Here,
$$f(x+5)\leq f(x+4) +1,$$
I didn't get any idea..
| Rewriting the given inequalities in terms of $\,f(x)=g(x)+x-1\,$:
$$\require{cancel}
\begin{align}
f(x+5)\geq f(x) + 5 \quad&\iff\quad g(x+5)+\bcancel{x}+\cancel{5}- \xcancel{1} \ge g(x) + \bcancel{x} -\xcancel{1} + \cancel{5} \\
&\iff\quad g(x+5) \ge g(x) \\
f(x+1)\leq f(x)+1 \quad&\iff\quad g(x+1)+\bcancel{x}+\cance... | {
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"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Proof of inequalities using binomial theorem
How do they achieve $$1+1+\frac{1}{2}\left(1-\frac{1}{n}\right)?$$ If I consider the first three terms of the binomial expansion, I get the first two terms, but can't factor the third to match this.
| $(1+x)^n = 1 + n x + \frac {n(n-1)}{2} x^2 + \cdots$
All the terms in the "$\cdots$" are positive if $x$ is positive.
If $n \ge 2$
$(1+x)^n \ge 1 + n x + \frac {n(n-1)}{2} x^2 $
$(1+\frac {1}{n})^n$ replace $x$ above with $\frac 1n$
$(1+\frac 1n)^n \ge 1 + n \frac 1n + \frac {n(n-1)}{2} (\frac 1n)^2 $
and distribute
$(... | {
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"url": "https://math.stackexchange.com/questions/2639365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Determine the whether the sequence $a_{n+1} = 3 - \frac{1}{a_n} \text{ for n > 1}$ is convergent or divergent. Consider the recursively defined sequence $a_n = 1$
$$a_{n+1} = 3 - \frac{1}{a_n} \text{ for n > 1}$$
Is the sequence convergent?
This is my attempt:
First, we prove that the sequence is positive, and monotoni... | A shorter approach, look at the function $f(x)=3-\frac{1}{x}$ because this function "generates" the sequence, i.e. $f(a_n)=a_{n+1}$. This function is ascending, because $f'(x)=\frac{1}{x^2}$ and $\frac{1}{3}<a_1<a_2 \Rightarrow 0< \color{red}{f(a_1)\leq f(a_2)} \Rightarrow \frac{1}{3}<a_1<\color{red}{a_2 \leq a_3}$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2639499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A triangle has the maximum possible area on the parabola $y = x^2$ Given points $P(−1, 1)$ and $Q(2, 4)$ on the parabola $y = x^2$, where should the point $R$ be on the parabola (between P and Q) so that the triangle $PQR$ has the maximum possible area?
How should I do this question?
| Let $R(x,x^2),$ where $-1\leq x\leq2.$
We have
$$m_{PQ}=1,$$ which gives the equation of $PQ:$
$$y-1=1(x+1)$$ or $$x-y+2=0.$$
Thus, the altitude of the triangle from $R$ is
$$\frac{|x-x^2+2|}{\sqrt2}$$ and since $$PQ=\sqrt{3^2+3^2}=\sqrt{18},$$ by AM-GM we obtain:
$$S_{\Delta PQR}=\frac{\sqrt{18}\cdot\frac{|x-x^2+2|}{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
} |
Determinant of a matrix after changes If $\det \begin{pmatrix}a&1&d\\ b&1&e\\ c&1&f\end{pmatrix}=1$ and $\det \begin{pmatrix}a&1&d\\ b&2&e\\ c&3&f\end{pmatrix}=1$, what is $\det \begin{pmatrix}a&-4&d\\ b&-5&e\\ c&-6&f\end{pmatrix}$?
So I am aware about all the different operations and what changes they bring to the val... | By the multilinearity of determinant,
$$\begin{aligned}
& \begin{vmatrix}a&-4&d\\ b&-5&e\\ c&-6&f\end{vmatrix} \\
&= \begin{vmatrix}a&-1&d\\ b&-2&e\\ c&-3&f\end{vmatrix}+\begin{vmatrix}a&-3&d\\ b&-3&e\\ c&-3&f\end{vmatrix} \\
&= -\begin{vmatrix}a&1&d\\ b&2&e\\ c&3&f\end{vmatrix}-3\begin{vmatrix}a&1&d\\ b&1&e\\ c&1&f\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Solve this initial value problem $y'=\tan^2(5x+5y)$, with the initial condition $y(0)=0$
So I'm about to lose my hair, I already have 60 attempts done for this problem.
I took $u= 5x+5y$, which gave me $u'= 5+5y'$.
Then I isolated $y'= (u'-5)/5$, which would give me
$5\tan^2(u)+5 = u'$ and then I have integral of $d... | Let us observe that
\begin{align}
y' =\tan^2(5x+5y) \ \ \implies& \ \ \cos^2(5x+5y)y' = \sin^2(5x+5y) = 1-\cos^2(5x+5y)\\
\implies& \ \ \cos^2(5x+5y)(1+y') = 1.
\end{align}
Next, notice if we define $u(x) =5y(x)+5x$, then it follows
\begin{align}
\cos^2(5x+5y)(1+y') = 1 \ \ \implies \ \ \cos^2(u)u' = 5 \ \ \implies \ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Congruence equation doubt Given the congruence equation:
$$70x\equiv 36 \pmod{198}$$
I have to find the smallest positive solution. I solved this by attempts (dividing the multiples of $70$ by $198$ and checking if the remainder is $36$) and I've got $x=9$ as a solution. Is my method incorrect?
| $$70x-36 = 198k$$
Divide it $2$,
$$35x-18=99k$$
We want to find the smallest positive integer $x$ such that an integer $k$ exists.
I would be great if we can find $35^{-1} \pmod{99}$, we know it exists since $35$ and $99$ are corprime.
Let's use Euclidean algorithm:
\begin{align}99&=2(35)+29\\
35&=1(29)+6\\
29&=4(6)+5\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Proving $ \frac {(a+b)^2 }{\sqrt{(a+b)^2 +1}} < 2 \left( \frac {a^2 }{\sqrt{ a^2 +1}} + \frac{b^2 } {\sqrt{ b^2 +1 }} \right) $
For positive real $a, b$, $$ \frac {(a+b)^2 }{\sqrt{(a+b)^2 +1}} < 2 \left( \frac {a^2 }{\sqrt{ a^2 +1}} + \frac{b^2 } {\sqrt{ b^2 +1 }} \right).$$
I know the inequality $ (a+b)^2 \le 2... | By C-S we obtain:
$$2\left(\frac {a^2}{\sqrt{a^2 + 1}} + \frac{b^2}{\sqrt{b^2 + 1}}\right)\geq\frac{2(a+b)^2}{\sqrt{a^2+1}+\sqrt{b^2+1}}\geq$$
$$\geq\frac{2(a+b)^2}{\sqrt{(1^2+1^2)(a^2+1+b^2+1)}}=\frac{(a+b)^2}{\sqrt{\frac{a^2+b^2}{2}+1}}\geq\frac{(a+b)^2}{\sqrt{(a+b)^2+1}}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2648806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
On the conjecture that if $\,\forall\{x, y, z\}\subset \Bbb{Z^+}$ squarefree, and $\ x^3 + y^3 = z^2$, then $\gcd(3, x, y) = 1$. I made a conjecture that if $\,\forall\{x, y, z\}\subset\mathbb{Z}^+$ with $x, y, z$ squarefree (not raised to a power greater than $1$; not a perfect power), and $$x^3 + y^3 = z^2,$$ then $\... | As stated your conjecture is false.
I supposed that $x,y$ are squarefree integers.
The following paramertization holds,
$$ x = s^{4} + 6 \, s^{2} t^{2} - 3 \, t^{4}, \
y = -s^{4} + 6 \, s^{2} t^{2} + 3 \, t^{4},\
z = 6 \, {\left(s^{4} + 3 \, t^{4}\right)} s t$$
(See for instance lemma 3.2.6 of Zagier http://www.ce... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2651638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to calculate the integral $\int\frac{1}{\sqrt{(x^2+8)^3}}dx$? I need to solve something like this
$$\int\frac{1}{\sqrt{(x^2+8)^3}}dx$$
Wolfram alpha says the solution is $$\frac{x}{8\sqrt{x^2+8}} + c$$
The problem is that the integrand is obtained by the quotient rule:
$$\bigg(\frac{g(x)}{h(x)}\bigg)'=\frac{g'(x)h(... |
It there a way to extract the solution from these types of integrals which argument is born from the easy quotient rule?
None that I know of. You just see it or maybe make a lucky substitution.
Alternatively, you can take a look at Euler substitutions, which will work in this case and can help for quite a lot of han... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2653216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Inductive proof of sequence defined by recurrence relation I'm working on a proof that shows for any $a \in \mathbb Q$ such that $a < 2$, there exists some $b \in \mathbb Q$ such that $a < b^2 < 2$. It was suggested that as part of the proof, I should consider the series defined by $a_1=1, a_{n+1}=\frac{2a_n+2}{a_n+2}$... |
$2-a_{k+1}^2 = \dfrac{-2a_k^2+4}{(a_k+2)^2}$
You are essentially done at this point. All that's left to note is that $\,a_k \ge 0\,$, and therefore:
$$
\\2-a_{k+1}^2 \,=\, \frac{2}{(a_k+2)^2} \cdot (2-a_k^2) \;\le\; \frac{2}{2^2} \cdot \frac{1}{k} \;\le\; \frac{1}{k+1}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2654945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How do I find the singular solution of the differential equation $y' = \frac{y^2 + 1}{xy + y}$? I start out with the separable differential equation,
$$y' =\frac{dy}{dx} = \frac{y^2 + 1}{xy + y} = \frac{y^2 + 1}{y(x+1)}$$
Thus, $\frac{1 }{x+1}dx = \frac{y }{y^2 + 1}dy$.
Then integrating both sides of the equation, I g... | You should get $x+1=C_1\sqrt{y^2+1}$, where $C_1=e^C$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2655177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
When is the following matrix diagonalizable? $\begin{pmatrix}8&k\\ 0&4\end{pmatrix}$
So I know that for a matrix to be diagonalizable, it needs to have 2 distinct real eigenvalues.
So I calculated the characteristic polynomial to be:
$x^2-12x+32-k$
Therefore, for the discriminant to be greater than $0$, I got the ineq... | It is more concrete to just go ahead and see what happens. With the eigenvalues $8,4$ we can get a matrix with eigenvectors as columns from
$$
R =
\left(
\begin{array}{rr}
1 & -k \\
0 & 4
\end{array}
\right).
$$
We can confirm this with
$$
\left(
\begin{array}{rr}
8 & k \\
0 & 4
\end{array}
\right)
\left(
\begin{ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2657633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to extract $(x+y+z)$ or $xyz$ from the determinant Prove
$$\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... | The determinant must be a polynomial in $x,y,z$ of degree $6$.
If you set $x=0$,
$$
\begin{vmatrix}
(y+z)^2&0&0\\
0&z^2&yz\\
0&yz&y^2
\end{vmatrix}=0
$$
so that $x$ is a factor. And by symmetry, $xyz$ as well.
Then with $x=-y-z$,
$$
\begin{vmatrix}
(y+z)^2&xy&zx\\
xy&y^2&yz\\
xz&yz&z^2
\end{vmatrix}=0
$$ (the bottom si... | {
"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": 0
} |
Find the kernel of a 4x4 matrix $$
\begin{pmatrix}
1 & 2 & 3 & 4\\
5 & 6 & 7 & 8\\
9 & 10 & 11 & 12\\
13 & 14 & 15 & 16\\
\end{pmatrix}
$$
I am asked to find the kernel of the matrix $M$. After doing some row operation I get to
$$
\begin{pmatrix}
1 & 2 & 3 & 4\\
0 & -4 & -8 & -12\\
... | Yes it looks good great. However, you should precise why the dimension of Kernel is $2$. For example by extracting the $2 \times 2$ determinant
$$
\begin{vmatrix}
3&4 \\
7&8
\end{vmatrix}=24-28 \ne 0
$$
So the dimesion of the image is at least $2$. Then you found two vectors in it, so the dimension of the image can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2661571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Find $\cos2B+\cos2C$ in a triangle In triangle $ABC$ we have $2a^2=b^2+c^2$. The measure of angle $A$ is $30^{\circ}$. Find $\cos2B+\cos2C$
| By the law of cosines we obtain:
$$a^2=b^2+c^2-2bc\cdot
\frac{\sqrt3}{2},$$
which gives $$2(b^2+c^2-\sqrt3bc)=b^2+c^2$$ or $$b^2+c^2=2\sqrt3bc.$$
Also, we have $$a^2=\sqrt3bc.$$
Thus, $$\cos2\beta+\cos2\gamma=2\left(\frac{a^2+c^2-b^2}{2ac}\right)^2+2\left(\frac{a^2+b^2-c^2}{2ab}\right)^2-2=$$
$$=2\left(\frac{\frac{b^2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $n^2 - 2n + 7$ is even then $n + 1$ is even I have to use Proof by contradiction to show what if $n^2 - 2n + 7$ is even then $n + 1$ is even.
Assume $n^2 - 2n + 7$ is even then $n + 1$ is odd. By definition of odd integers, we have $n = 2k+1$.
What I have done so far:
\begin{align}
& n + 1 = (2k+1)^2 - 2(2... | $n^2-2n+7
=n^2+2n+1-(4n+6)
=(n+1)^2-2(2n+3)
$.
If this is even then,
since $2(2n+3)$ is even,
their sum is even,
so $(n+1)^2$ is even
so $n+1$ is even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2662554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 5
} |
Compute the volume bounded by the parabolic cylinders $x^2=4-4z, \quad y^2=4-4z$ and the $xy-$plane.
Compute the volume bounded by the parabolic cylinders $x^2=4-4z, \quad
> y^2=4-4z$ and the $xy-$plane.
Say the $xy-$plane is the ground, the two cylindrcal paraboloids then make upp the roof, where the base is simply ... | Note that in the integral $x$ varies between $0$ and $2$, thus
$$8\int_0^2 x-\frac{x^3}{4} \ dx = \left[\frac{x^2}2-\frac{x^4}{16}\right]_0^2=8(2-1)=8$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Diophantine Equation $a^6+b^6+c^6=d^6$ Within the literature on Diophantine equations there seems to be very little on the $6,1,3$ equation:
$$a^6+b^6+c^6=d^6\quad\quad(1)$$
Mathworld, for example, simply records that there are no known solutions of $6.1.n$ equations for $n \leq 6$. The former Euler Project searching f... | COMMENT (This is not an answer!)
The identity $(2xz)^2+(2yz)^2+(z^2-x^2-y^2)^2=(x^2+y^2+z^2)^2$ gives the general solution of $X^2+Y^2+Z^2=W^2$ so one has the necessary condition with parameters $x,y,z$
$$\begin{cases}a^3=2xz\\b^3=2yz\\c^3=z^2-x^2-y^2\\d^3=x^2+y^2+z^2\end{cases}$$ This could be a good first step to get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2663646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Solve the following differential equations 2x2 with complex eigenvert
Solve the following equation:
$$X'=\pmatrix { 2 & 5
\\ -5 & 8}X$$
$$x(0)=(5,5)^t$$
I already got out the followings:
$$\lambda_{1,2} = 5 \pm 4i$$
$$v_{1} = \begin{pmatrix} 3 \\ 5 \end{pmatrix} - i\cdot \begin{pmatrix} 4 \\ 0 \end{pmatrix}$$
... | Since we have complex conjugate eigenvalues/eigenvectors, we only need to use one of them.
For $\lambda_1 = 5 + 4 i, v_1 = \begin{pmatrix} 3 -4i\\ 5 \end{pmatrix} $, we have
$\begin{align} e^{(5 + 4i)t} \begin{pmatrix} 3 -4i\\ 5 \end{pmatrix} \\ = e^{5t}e^{4it} \begin{pmatrix} 3 -4i\\ 5 \end{pmatrix}\end{align} \\
= e^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Calculating the summation $\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}$ I need to find explicitly the following summation
$$\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}, \quad
H_{n}=1+\frac{1}{2}+\cdots+\frac{1}{n}$$
From Mathematica, I checked that the answer is $2$. The same result is returned by WolframAlpha.
A thought ... | (Big) hint:
$$\frac{H_{n+1}}{n(n+1)}
= \frac{H_{n+1}}{n}-\frac{H_{n+1}}{n+1}
= \frac{1}{n(n+1)} + \frac{H_{n}}{n}-\frac{H_{n+1}}{n+1}$$
so that you can get a telescopic series: for any $N\geq 1$,
$$
\sum_{n=1}^N \frac{H_{n+1}}{n(n+1)}
= \sum_{n=1}^N \frac{1}{n(n+1)} + \sum_{n=1}^N \frac{H_{n}}{n} - \sum_{n=2}^{N +1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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real value of $k$ inirrational equation Find value of $k$ for which the equation $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}$ has no solution.
solution i try $\sqrt{3z+3}-\sqrt{3z-9}=\sqrt{2z+k}......(1)$
$\displaystyle \sqrt{3z+3}+\sqrt{3z-9}=\frac{12}{\sqrt{2z+k}}........(2)$
$\displaystyle 2\sqrt{3z+3}=\frac{12}{\sqrt{2z+k... | Starting from your last step,
$$4(6z^2+6z+3kz+3k)=144+4z^2+k^2+48z+4kz+24k$$
is equivalent to
$$28z^2+(24+12k+48+4k)z-(k^2+12k-144)=0 $$
and this as no solution (since $z$ is real, for the squared roots be defined) if and only if
$$ \Delta = (24+12k+48+4k)^2-4\times 28 \times -(k^2+12k-144) <0 $$
So it remain to study... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2665723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\frac{1}{a^n+b^n+c^n} = \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}$ Let $n \in N, n=2k+1, and \text{ } \frac{1}{a+b+c} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$.
Show that $$\frac{1}{a^n+b^n+c^n} = \frac{1}{a^n} + \frac{1}{b^n} + \frac{1}{c^n}$$
I have tried, but I don't get anything. Can you please gi... | Hint
*
*At the first step, show that $(a+b)(a+c)(b+c)=0$
*Next, show that two of the three numbers are opposite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2668597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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$36 \leq 4(a^3+b^3+c^3+d^3) - (a^4+b^4+c^4+d^4)\leq4 8.$ Let $a,b,c,d \in \Bbb R$, $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12$. Then
$$36 \leq 4(a^3+b^3+c^3+d^3) - (a^4+b^4+c^4+d^4)\leq4
8.$$
I have found only two bounds: $216 \geq a^3+b^3+c^3+d^3$ and $144 \geq a^4+b^4+c^4+d^4$.
How to prove this inequality?
| For upper bound, we have
$$ x^3(4-x) \leq 4x^2\;\; \forall x$$
and we are done.
For lower bound:
Let $a+b+c =x$, then $d=6-x$ and by Cauchy we have $a^2+b^2+c^2\geq x^2/3$
so $$12 \geq {x^2\over 3} +(6-x)^2$$ so $x\in (3,6) $ and so $0\leq d\leq 3$. But this holds also for $a,b,c$.
Idea for finish:
Let $f(x) = 4x^3... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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How to calculate $ \int \frac{\sin^{6}(x)}{\sin^{6}(x) + \cos^{6}(x)} dx? $ How to calculate
$$ \int \frac{\sin^{6}(x)}{\sin^{6}(x) + \cos^{6}(x)} dx? $$
I already know one possible way, that is by :
$$ \int \frac{\sin^{6}(x)}{\sin^{6}(x) + \cos^{6}(x)} dx = \int 1 - \frac{\cos^{6}(x)}{\sin^{6}(x) + \cos^{6}(x)} dx $... | An alternative would be to do a load of trig manipulation:
Starting with $$\frac{s^6}{s^6+c^6}=1-\frac{c^6}{s^6+c^6}$$
$$=1-\frac{\frac 18(1+\cos2x))^3}{ \frac 18(1+\cos2x))^3+\frac 18(1-\cos2x))^3}$$
= a few more lines of algebra = $$=\frac 12+\frac 12\cos 2x\left(\frac{3+\cos^2 2x}{1+3\cos^2 2x}\right)$$
$$=\frac 12... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2671984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
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Find a pattern and prove it by mathematical induction: $1 = 1$
$3 + 5 = 8$
$7 + 9 + 11 = 27$
$13 + 15 + 17 + 19 = 64$
Etc...
I am having trouble seeing a pattern with this, I know it is relatable with Fibonacci Numbers but I am having trouble grasping this topic
| Let $a_n$ the $n$-th term of the sequence.
The sum of the $n$ first terms is the sum of the first $\frac{n(n+1)}{2}$ odd integers, which is $\left( \frac{n(n+1)}{2} \right)^2$.
So that:
\begin{equation}
a_1+\ldots +a_n = \left( \frac{n(n+1)}{2} \right)^2
\end{equation}
\begin{equation}
a_1+\ldots +a_{n-1} = \left( \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2672703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
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Sketch the region in the plane consisting of all points (x,y) such that $2xy\le |x-y|\le x^2+y^2$ Question: Sketch the region in the plane consisting of all points (x,y) such that $2xy\le |x-y|\le x^2+y^2$
My attempt:
If $x>y$ then
$|x-y|=x-y$
Thus now,
$2xy\le x-y\le x^2+y^2$
If $x-y\geq 2xy$ then let $x-y=k$ ($k$ is ... | we have $$2xy\le |x-y|\le x^2+y^2$$ now we assume $$x\geq y$$ then we get
$$2xy\le x-y$$ and $$x-y\le x^2+y^2$$
the right inequality Can be written as $$\frac{1}{2}\le (x-\frac{1}{2})^2+(y-\frac{1}{2})^2$$
the left Hand side is:
$$y(1+2x)\le x$$
$$x=-\frac{1}{2}$$ gives a contradiction
if $$x<-\frac{1}{2}$$ we get $$y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2675028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Greatest Common Divisor of two Polynomials. Find the $\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$ over $\mathbb{Q}[x]$. Then find two polynomials $a(x),b(x) \in \mathbb{Q}[x]$ such that, $$a(x)(x^3-6x^2+14x-15) + b(x)(x^3-8x^2+21x-18)=\gcd(x^3-6x^2+14x-15, x^3-8x^2+21x-18)$$
I have managed to find,
$$x^3-6x^2+14x-15=(x-... | This is just the Extended Euclidean Algorithm. Instead of back-substitution, I have always preferred to write the construction steps in the style of continued fractions. Furthermore, I have always depended on the kindness of strangers.
$$ \left( x^{3} - 6 x^{2} + 14 x - 15 \right) $$
$$ \left( x^{3} - 8 x^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
$\lim_{x\rightarrow 0} \frac{2\cdot \ln(1+x)+x^2-2x}{x^3}$ without 'Hopital Rule Compute $$\lim_{x\rightarrow 0} \frac{(x+2)\cdot \ln(1+x)-2x}{x^3}$$ without L'Hopital Rule.
I proved before that that $$\lim_{x\rightarrow 0} \frac{ \ln(1+x)-x}{x^2}=-\frac{1}{2}$$
I tried to use it and I have to compute $$\lim_{x\righta... | As shown, without l'Hopital, the limit can be solved easily by Taylor's expansion that is given the two following limits
*
*$\frac{\ln(1+x)-x}{x^2}\to -\frac12$ (as already mentioned)
and
*
*$\frac{\ln(1+x)-x+\frac{x^2}2}{x^3}\to \frac13$ (which correspond to to your second limit)
indeed by those limits, as y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2679777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Simplification algebraic of a cube root. I am trying to simplify this:
$$\frac{1000}{\pi \cdot (\frac{500}{\pi})^{\frac{2}{3}}}$$
and I think it becomes:
$$2 \cdot \sqrt[3]{\frac{500}{\pi}}$$
I basically thought we cube root the $\frac{500}{\pi}$ and then multiply the denominator by $\frac{500}{\pi}$ which could cancel... | $$\frac{1000}{ \pi \cdot (\frac{500}{\pi})^{\frac{2}{3}} }
=\frac{1000}{\pi^{\frac{1}{3}}\cdot 2^{-\frac{2}{3}}\cdot 2^{\frac{2}{3}}\cdot 500^{\frac{2}{3}}}
=\frac{1000}{\pi^{\frac{1}{3}}\cdot 2^{-\frac{2}{3}}\cdot 1000^{\frac{2}{3}}}
=\frac{2\cdot 1000^{\frac{1}{3}}}{(2\pi)^{\frac{1}{3}}}
=\frac{20}{\sqrt[3] {2\pi} }=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find $\int_0^{\sqrt{R^2-1}}g(x)-f(x)\,\mathrm dx$
How do you evaluate this integral?
$$\int_0^{\sqrt{R^2-1}}g(x)-f(x)\,\mathrm dx$$
where $f(x)=1$ and $g(x)=\sqrt{R^2-x^2}$.
Wolfram tells me I exceeded my computational limit. Mathematica gives me a long answer which is very difficult to read (for me).
EDIT: I was... | Using linearity, you can split the integral
$$\begin{align*}\int_{0}^{\sqrt{R^2 - 1}} g(x) - f(x)~\mathrm{d}x &= \int_{0}^{\sqrt{R^2 - 1}} \sqrt{R^2 - x^2} - 1~\mathrm{d}x \\ &= \int_{0}^{\sqrt{R^2 - 1}} \sqrt{R^2 - x^2}~\mathrm{d}x - \int_{0}^{\sqrt{R^2 - 1}} 1~\mathrm{d}x\tag{1}\end{align*}$$
Let's first solve the fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683423",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Area of largest inscribed rectangle in an ellipse. Can I take the square of the area to simplify calculations? So say I have an ellipse defined like this:
$$\frac{x^2}{9} + \frac{y^2}{4} = 1$$
I have to find the largest possible area of an inscribed rectangle.
So the area ($A$) of a rectangle is $2x2y=4xy$. Also we can... | I agree with José Carlos Santos, yet I don't agree with your $A^2$ derivative. So, I like to show my way of solving with $A^2$.
$$Area =A= 4x \cdot \left(4 - \frac{4x^2}{9}\right)^{\frac12}$$
$$A^2= 16x^2 \cdot \left(4 - \frac{4x^2}{9}\right) = 64x^2 - \frac{64x^4}{9}$$
Thus, derivative of $A^2$: $$ \frac1A\cdot\frac{d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2685225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 3
} |
On the series $\sum \limits_{n=1}^{\infty} \frac{1}{n^2-3n+3}$ and $\sum\limits_{n=-\infty}^{\infty} \frac{1}{n^2-3n+3}$ Wolfram Alpha says that
$$\sum_{n=1}^{\infty} \frac{1}{n^2-3n+3} = 1 + \frac{\pi \tanh \left ( \frac{\sqrt{3}\pi}{2} \right )}{\sqrt{3}}$$
However I am unable to get it. It is fairly routine to prove... | $$\sum_{n\geq 0}\frac{1}{(n+a)(n+b)}=\frac{\psi(a)-\psi(b)}{a-b}\tag{1}$$
for any $a\neq b$ in the half-plane $\text{Re}(s)>0$ is fairly routine, too. Here we have to find
$$ \sum_{n\geq 0}\frac{1}{n^2-n+1}=1+\sum_{n\geq 0}\frac{1}{n^2+n+1}=1+\frac{\psi\left(\frac{1+i\sqrt{3}}{2}\right)-\psi\left(\frac{1-i\sqrt{3}}{2}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2686931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Hyperbolic functions proof using $\cosh x$
Prove that $\cosh x-1\equiv \frac12(e^{0.5x}-e^{-0.5x})^2$
I'm stuck on what appears to be the last step, please could someone explain where I have made a mistake?
\begin{align}
\frac12(e^{0.5x}-e^{-0.5x})^2 & \equiv \frac12(\sinh(0.5x))^2 \\
& \equiv \frac12\sinh^2(0.5x) \\... | Note that $$\begin{align}
\frac12(e^{0.5x}-e^{-0.5x})^2 & \equiv \frac12(2\sinh(0.5x))^2 \\
& \equiv 2\sinh^2(0.5x) \\
& \equiv 2(\cosh^2(0.5x) -1) \\
& \equiv 2((\frac{e^{0.5x}+e^{-0.5x}}{2})^2 -1) \\
& \equiv 2((\frac{e^x+e^{-x}+2}4) -1) \\
& \equiv ((\frac{e^x+e^{-x}+2}2)-2) \\
& \equiv (\frac{e^x+e^{-x}}2+\frac2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2689521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"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.