Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Find the intersection points of two circles Find the intersection points of the circles $$k_1:(x-4)^2+(y-1)^2=9\\k_2:(x-8)^2+(y+4)^2=100$$
The intersections point (if they exist) will satisfy the equations of both the circles, so we can find their coordinates by solving the system $$\begin{cases}(x-4)^2+(y-1)^2=9\\(x-... | If you're familiar with implicit differentiation, we can locate the points where the slopes of the tangent lines to the two circles "match". We have
$$ (x \ - \ 4)^2 \ + \ (y \ - \ 1)^2 \ \ = \ \ 9 \ \ \Rightarrow \ \ 2·(x \ - \ 4) \ + \ 2·(y \ - \ 1)·y' \ \ = \ \ 0 \ \ \Rightarrow \ \ y' \ \ = \ \ \frac{4 \ - \ x}{y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4618115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Find $\lim_{x\to +\infty}\frac{\sqrt {2x^2+3}}{4x+2}$ and $\lim_{x\to -\infty}\frac{\sqrt {2x^2+3}}{4x+2}$. I was reading about limits of functions from Problems of Calculus in One Variable written by I.A Maron. There was an example given in the book:
Find $\lim_{x\to +\infty}\frac{\sqrt {2x^2+3}}{4x+2}$ and $\lim_{x\t... | $$
\text { Noting that } \sqrt{x^2}=\left\{\begin{array}{cl}
x & \text { if } x \geq 0 \\
-x & \text { if } x<0
\end{array}\right.
$$
$$
\begin{aligned}
\lim _{x \rightarrow+\infty} \frac{\sqrt{2 x^2+3}}{4 x+2} & =\lim _{x \rightarrow+\infty} \frac{\frac{1}{\sqrt{x^2}} \sqrt{2 x^2+3}}{4+\frac{2}{x}} \\
& =\lim _{x \rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4620744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$\lim_{(x,y)\to (0,0)}\frac{1-(\cos x)(\cos y)}{x^2+y^2} $ I need to find the limit for $$\lim_{(x,y)\to (0,0)}\frac{1-(\cos x)(\cos y)}{x^2+y^2} $$
whether exist.
I use many example (ex:line, interated limit, half angle formula, ...), and I always get the answer $1/2$. However, this does not mean the limit is $1/2$. T... | Another way to calculate the limit
$\lim\limits_{(x,y)\to(0,0)}\dfrac{1-\cos x\cos y}{x^2+y^2}\;.$
First of all, we will calculate the following limit :
$\lim\limits_{(u,v)\to(0,0)}\dfrac{\sin^2\!u+\sin^2\!v}{u^2+v^2}\;.$
Let $\;\varphi:\,]\!-\!\infty,+\infty[\to\Bbb R\;$ be the function defined as :
$\varphi(t)=\begin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4621053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Calculating $\displaystyle{\lim_{x \to 0^+}}{\frac{1}{\sqrt{x}}\Big(e^x + \frac{2\log(\cos(x))}{x^2}}\Big)$ I am struggling to calculate this limit:
$$\displaystyle{\lim_{x \to 0^+}}{\frac{e^x + \frac{2\log(\cos(x))}{x^2}}{\sqrt{x}}}$$
I prefer not to use l'Hopital's rule, only when necessary. If possible, solving with... | As the OP asks, an attempt to do it, using only
\begin{align}
\log(1+u)&=u+o(u),\\
e^x &= 1 + x + o(x),\\
\cos(x) &= 1 -\frac{x^2}{2} + o(x^2) .
\end{align}
Now, as $x \to 0$, the best we can deduce is:
\begin{align}
\cos x &= 1 - \frac{x^2}{2} + o(x^2)
\\
\log \cos x &= \log(1-(1-\cos x)) = -(1-\cos x) + o\big(1-\cos ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4622468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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prove $(\sin x)^{-2}-x^{-2}\leq 1-\frac{4}{{\pi}^{2}},x\in(0,\pi/2]$ $(\sin x)^{-2}-x^{-2}\leq 1-\frac{4}{{\pi}^{2}},x\in(0,\pi/2]$
How to deal with this problem?
Observing that when $x=\pi/2$, the above inequality becomes equality.
Firstly, denote $f(x)=(\sin x)^{-2}-x^{-2}$ and then take derivative of $f(x)$. We have... | The Laurent series expansion of $\frac{1}{\sin x} $, valid for $0<|x|< \pi$, is
$$\frac{1}{\sin x} = \frac{1}{x} + \frac{1}{6} x + \frac{7}{360} x^3 + \frac{31}{1520} x^5 + \cdots $$
with all coefficients positive. We conclude that also $\frac{1}{\sin^2 x}$ has a Laurent expansion with positive coefficients valid for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4626032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove $\frac{ab^2+2}{a+c} +\frac{bc^2+2}{b+a} +\frac{ca^2+2}{c+b} \geq \frac{9}{2}$ for $a,b,c\geq1$ Prove $\dfrac{ab^2+2}{a+c} +\dfrac{bc^2+2}{b+a} +\dfrac{ca^2+2}{c+b} \geq \dfrac{9}{2}$ for $a,b,c\geq1$.
I tried using the Titu Andreescu form of the Cauchy Schwarz inequality and got to this point:
$\dfrac{(b\sqrt{a}+... | I will assume your deduction is correct and I will try to prove the inequality to which you have reduced the original inequality.
Consider the sequences
$$(c_i)_{i=1}^{3}:=(\sqrt{bc},\sqrt{ac},\sqrt{ab}),\;\;(d_{i})_{i=1}=(\sqrt{a},\sqrt{b},\sqrt{c})$$
Notice that for any permutation $\sigma:\{1,2,3\}\to\{1,2,3\}$ if
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4628757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 1
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Sine curve that passes through (1,1) (2,2) and (3,3) Question: Find a sine function in the form $a\sin(bx) + c$ that passes through points (1,1) (2,2) and (3,3)
Working so far:
*
*We have three points for three unknown variables in the function, so we can use simultaneous equations to solve for them.
*Simultaneous ... | The solutions are
$$ f(x) = -\sin\left(\left(\frac{\pi}{2} + 2\pi k\right)x\right) + 2$$
and
$$ f(x) = \sin\left(\left(\frac{3\pi}{2} + 2\pi k\right)x\right) + 2$$
We can add the first and third equations to get
$$a\ \sin(b) + a\ \sin(3b)+2c=4 \\ a\ \sin(2b-b)+a\ \sin(2b+B)+2c=4 \\ 2a\ \sin(2b)\cos(b) + 2c = 4 \\a\ \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4630346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Evaluate $\sum_{r=1}^{\infty} \dfrac{r^2 - 1}{r^4 + r^2 + 1}$ I was only able to observe that:
$\dfrac{r^2 - 1}{r^4 + r^2 + 1} = \dfrac{r^2 - 1}{(r^2 + r + 1)(r^2 - r + 1)}$
This hints at telescoping, but I would need an $r$ term in the numerator.
The original question was
Evaluate $\sum_{r=1}^{\infty} \dfrac{r^3 + (... | If you are comfortable with generalized harmonic number, you could consider first the partial sum
$$S_n=\sum_{r=1}^{n} \dfrac{r^2 - 1}{r^4 + r^2 + 1}$$ and write first
$$\dfrac{r^2 - 1}{r^4 + r^2 + 1}=\frac{(r-1)(r+1)}{(r-a)(r-b)(r-c)(r-d)}$$
Using partial fraction decomposition, this is
$$\frac{a^2-1}{(a-b) (a-c) (a-d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4631515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 0
} |
Bijective mapping/substitution - proof? Consider the ellipse
$$x^2/a^2 + y^2/b^2 = 1$$
(as a curve $L$) and its inner part $B$.
So $B$ is defined as
$$x^2/a^2 + y^2/b^2 < 1$$
Here $a,b$ are positive constants.
Both $L, B \subseteq \mathbb{R}^2$
Then consider just the inner part of this ellipse $B$ (excluding the contou... | I can help you with injectivity, not yet sure about surjectivity.
Consider $(r_1,\theta_1) \neq (r_2,\theta_2) $ and $ f(r_1,\theta_1) = f(r_2,\theta_2) $, or,
$$\begin{align*}
(a \cdot r_1 \cos \theta_1, b\cdot r_1 \sin \theta_1) &= (a\cdot r_2 \cos \theta_2, b \cdot r_2 \sin \theta_2)
\end{align*}$$
This gives us t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4644512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Product of limit, sin, infinity, error? Hello I would like to know if there is a mistake :
I have to show that for any $t\geqslant0$ fixed
$$\lim_{n\to \infty}\sin\sqrt{t+4\pi n^{2}}=0$$
That's what I said,
Since $\sin(\cdot)$ is continuous and $$\sqrt{t+4\pi n^{2}}=2n\pi\sqrt{1+\frac{t}{4\pi n^{2}}}$$ for $n\geqslant1... | Actually it does not exist the limit:$$\lim_{n\to \infty}\sin\sqrt{t+4\pi n^{2}}\;.$$
So, I think the OP intended to write the following limit:$$\lim_{n\to \infty}\sin\sqrt{t+4\pi^2n^{2}}\;.\quad(\text{ where }n\in\Bbb N\;)$$
Indeed ,
$\lim\limits_{n\to \infty}\sin\sqrt{t+4\pi^2n^{2}}=$
$=\lim\limits_{n\to \infty}\sin\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4645543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
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Investigate on the convergence of $I(a)= \int_0^{\infty} \frac{x}{\sqrt{1+x^a}} d x $ for $a>0$ and its exact value in case of convergence. Inspired by the post,
I start to investigate the convergence of $$I(a)= \int_0^{\infty} \frac{x}{\sqrt{1+x^a}} d x $$ for $a>0$.
For any $0\le a\le 4$, if $x\ge 1$, we have
$$
\f... | Similarly to the question you asked in, the Mellin transform of $f(x)=(1+x)^{-\rho}$ is
$$
\tilde{f}(s) = \mathcal{M}[f(x);s] = \frac{1}{\Gamma(\rho)}\Gamma(s)\Gamma(\rho-s)
$$
which is defined for $0<\Re(s)<\Re(\rho)$. Then, by the Mellin transform, one has $f(x^n) \rightarrow \frac{1}{|n|}\tilde{f}\left(\frac{s}{n} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4647411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Evaluate the infinite summation
Evaluate $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{n^2}{1+n^3}$$
I tried to factor the denominator and then using partial fraction $$\frac{n^2}{1+n^3}=\frac{n^2}{(n+1)(n^2-n+1)}$$ $$=\frac{2n-1}{3(n^2-n+1)}+\frac{1}{3(n+1)}$$
So our question now becomes $$\frac13\sum_{n=1}^{\infty}(-1)^{n+1... | Let $S$ be the sum with the quadratic in the denominator. As user Ron Gordon points out in the linked question,
$$\begin{align*}
S &= \sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2 k-1)}{k^2-k+1} \\
&= -\frac12 \sum_{k=-\infty}^{\infty} \frac{(-1)^{k} (2 k-1)}{k^2-k+1}
\end{align*}$$
Take out the $k=0$ term and split up the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4647976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Expected average of differences between two points in a uniform distribution Let's say I have a discrete uniform distribution where a variable x can take any value between 1 to 100 (inclusive).
I spin up two values of x: x1 and x2 and they take the values 23 and 53 respectively.
The absolute difference between x1 and x... | I started with two die $(n=6)$ and made a table $$\begin{array}{|c|c|c|c|c|c|}\hline \text{die 1 / die 2 } & 1 &2 &3 &4 &5 &6 \\ \hline\hline \hline 1 & 0&1 &2 &3 &4 &5 \\ \hline 2 & 1 &0 &1 &2 &3&4 \\ \hline 3 &2 &1 &0 &1&2&3 \\ \hline 4 &3 &2 &1&0&1&2 \\ \hline 5 &4 &3&2&1&0 &1 \\ \hline 6 &5&4&3&2 &1 &0 \\ \h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/4649004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $ I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,... | Grant Sanderson, aka 3Blue1Brown, has a good explanation of this in one of his Lockdown Math videos. His explanation, summarized:
*
*$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \ldots = f(1)$, where $f(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} -\ldots$
*$\frac{df}{dx} = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "60",
"answer_count": 12,
"answer_id": 9
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Is there a general formula for solving Quartic (Degree $4$) equations? There is a general formula for solving quadratic equations, namely the Quadratic Formula, or the Sridharacharya Formula:
$$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } $$
For cubic equations of the form $ax^3+bx^2+cx+d=0$, there is a set of three ... | We can reduce the problem of solving the general quartic to merely solving a quadratic. Given,
$$x^4+ax^3+bx^2+cx+d=0$$
Then the four roots are,
$$x_{1,2} = -\frac{a}{4}+\frac{\color{red}\pm\sqrt{u}}{2}\color{blue}+\frac{1}{4}\sqrt{3a^2-8b-4u+\frac{-a^3+4ab-8c}{\color{red}\pm\sqrt{u}}}\tag1$$
$$x_{3,4} = -\frac{a}{4}+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "165",
"answer_count": 12,
"answer_id": 4
} |
How closely can we estimate $\sum_{i=0}^n \sqrt{i}$ By looking at an integral and bounding the error?
| In case you were wondering, like me, Moron's excellent proof adapts easily to show that
$$1 + \sqrt[3]{2} + \dots + \sqrt[3]{n} \sim \frac{3}{4}n^{4/3} + \frac{\sqrt[3]{n}}{2} + C,$$
for some constant $C.$ In this case $C = \zeta(-1/3) \approx -0.277343.$
Where, as before, $a_n \sim b_n$ means $\lim_{n \rightarrow \inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/5676",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 7,
"answer_id": 0
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Evaluate $\int \frac{1}{\sin x\cos x} dx $ Question: How to evaluate $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\displaystyle\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However... | Integrand $ =\dfrac {1}{\sin(x)\cos(x)} = 2 \csc 2x $
Its integral is obtained by direct application of listed standard trigonometric function integration formulae. Using chain rule for constant double angle:
$$ 2 \log (\tan \dfrac{2 x}{2}) \cdot \frac12= \log (\tan x ) + c $$
agrees with OP's second result when it is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/9075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 10,
"answer_id": 7
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Floor of Square Root Summation problem I have problem calculating the following summation:
$$
S = \sum_{j=1}^{k^2-1} \lfloor \sqrt{j}\rfloor.
$$
As far as I understand the mean of that summation it will be something like $$1+1+1+2+2+2+2+2+3+3+3+3+3+3+3+\cdots$$
and I suspect that the last summation number will be $(k-1... | Since the last value for $j$ is $k^2-1$, none of the terms of the sum are $k$; they are all between $1$ and $k-1$.
How many $1$'s will be in the sum? Well, we'll get $1$ when $j$ is any number between $1^2$ and $2^2-1$; then we'll get $2$ for each number between $2^2$ and $3^2-1$. Then we'll get $3$ for each number bet... | {
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"url": "https://math.stackexchange.com/questions/10183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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What type of triangle satisfies: $\cot \biggl( \frac{A}{2} \biggr) = \frac{b+c}{a} $ If in a $\displaystyle\bigtriangleup$ ABC, $\displaystyle\cot \biggl( \frac{A}{2} \biggr) = \frac{b+c}{a} $, then $\displaystyle\bigtriangleup$ ABC is of which type ?
| So by Law of Sines we have $$ \frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}} =k (\text{say})$$
From this your equation becomes, $$\frac{\cos\frac{A}{2}}{\sin\frac{A}{2}} = \frac{k(\sin{B} + \sin{C})}{k \sin{A}} = \frac{\sin{B}+\sin{C}}{\sin{A}} = \frac{\cos\frac{A}{2} \cos\frac{B-C}{2}}{\sin\frac{A}{2} \cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/10545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Solving $\log _2(x-4) + \log _2(x+2) = 4$ Here is how I have worked it out so far:
$\log _2(x-4)+\log(x+2)=4$
$\log _2((x-4)(x+2)) = 4$
$(x-4)(x+2)=2^4$
$(x-4)(x+2)=16$
How do I proceed from here?
$x^2+2x-8 = 16$
$x^2+2x = 24$
$x(x+2) = 24$ Which I know is not the right answer
$x^2+2x-24 = 0$ Can't factor this
| It is $x^2-2x-8 = 16$ my friend. So you get $x^2 - 2x -24 = 0$, which factors as $(x-6)(x+4) = 0$. Hence, $x=6$ or $x = -4$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
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How to prove that: $\tan(3\pi/11) + 4\sin(2\pi/11) = \sqrt{11}$ How can we prove the following trigonometric identity?
$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
| Since $\tan\frac{3\pi}{11}+4\sin\frac{2\pi}{11}>0$, it's enough to prove that
$$\left(\sin\frac{3\pi}{11}+4\sin\frac{2\pi}{11}\cos\frac{3\pi}{11}\right)^2=11\cos^2\frac{3\pi}{11}$$ or
$$\left(\sin\frac{3\pi}{11}+2\sin\frac{5\pi}{11}-2\sin\frac{\pi}{11}\right)^2=11\cos^2\frac{3\pi}{11}$$ or
$$1-\cos\frac{6\pi}{11}+4-4\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/11246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 6,
"answer_id": 2
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Evaluation of the sum $\sum_{k = 0}^{\lfloor a/b \rfloor} \left \lfloor \frac{a - kb}{c} \right \rfloor$ Let $a, b$ and $c$ be positive integers. Recall that the greatest common divisor (gcd) function has the following representation:
\begin{eqnarray}
\textbf{gcd}(b,c) = 2 \sum_{k = 1}^{c- 1} \left \lfloor \frac{kb}{c}... | Here is an observation/partial result. For brevity write $t = \lfloor a/b \rfloor .$ When $\text{gcd}(b,c)=1$ and $c \, | \, (t+1) $ we have
$$ S = \sum_{k=0}^{t} \left \lfloor \frac{a - kb}{c} \right \rfloor =
\frac{t+1}{c} \left \lbrace a - \frac{tb}{2} - \frac{c-1}{2} \right \rbrace . $$
Proof:
Suppose
$$\begin{ali... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "12",
"answer_count": 2,
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If $\sin x + \cos x = \frac{\sqrt{3} + 1}{2}$ then $\tan x + \cot x=?$ Hello :)
I hit a problem.
If $\sin x + \cos x = \frac{\sqrt{3} + 1}{2}$, then how much is $\tan x + \cot x$?
| Another more general approach is to solve your equation for $x$. Since it is
linear in $\sin x$ and $\cos x$ it can be transformed into a quadratic
equation in $\tan \frac{x}{2}$ (see this answer):
$$\sin x+\cos x=\frac{1+\sqrt{3}}{2}\Leftrightarrow \frac{2\tan \frac{x}{2}}{%
1+\tan ^{2}\frac{x}{2}}+\frac{1-\tan ^{2}\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/19131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 2
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How to "Re-write completing the square": $x^2+x+1$ The exercise asks to "Re-write completing the square": $$x^2+x+1$$
The answer is: $$\left(x+\frac{1}{2}\right)^2+\frac{3}{4}$$
I don't even understand what it means with "Re-write completing the square"..
What's the steps to solve this?
| Remember the formula for the square of a binomial:
$$(a+b)^2 = a^2 + 2ab + b^2.$$
Now, when you see $x^2+x+1$, you want to think of $x^2+x$ as the first two terms you get in expanding the binomial $(x+c)^2$ for some $c$; that is,
$$x^2 + x + \cdots = (x+c)^2.$$
Since the middle term should be $2cx$, and you have $x$, t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the vertical asymptote of a function For an assignment, I was asked to find the vertical asymptote of the function $$g(x)= \frac{\frac{1}{2}x^3-4x^2+6x}{7x^2-56x+84}.$$
According to my text, a reliable method of finding the asymptote is to factor the numerator and denominator, and what left in the denominator tha... | Note that $\displaystyle f(x) = \frac{\frac{1}{2}x^3-4x^2+6x}{7x^2-56x+84} = \frac{1}{2}\frac{x^3-8x^2+12x}{7x^2-56x+84} = \frac{1}{2}\frac{1}{7}\frac{x^3-8x^2+12x}{x^2-8x+12} = \frac{x}{14}$.
So the function is "almost" a straight line passing through origin with a slope $\frac{1}{14}$ except at $x=2$ and $x=6$.
The f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/21778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Computing $\sum_{m \neq n} \frac{1}{n^2-m^2}$ A series arising in perturbation theory in quantum mechanics:
$\sum_{m\neq n} \frac{1}{n^2 - m^2}$, where $n$ is a given positive odd integer and $m$ runs through all odd positive integers different from $n$. I have a hunch that residue methods are applicable here, but I do... | You can write
$$ \frac{1}{n^2 - m^2} = \frac{1}{2n}
\left\lbrace \frac{1}{m+n} - \frac{1}{m-n} \right\rbrace . \quad (1)$$
Now if we sum up both sides over all odd $m \ne n ,$ taking into account that $n$ is odd, lots of cancelling goes on and we obtain
$$\sum_{m \ne n} \frac{1}{n^2 - m^2} = -\frac{1}{4n^2}.$$
At firs... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if $p^{a}$ is a factor of the canonical factorization of ${{2n}\choose{n}}$ then $p^{a} < 2n$? Prove that if $p^{a}$ is a factor of the canonical factorization of ${{2n}\choose{n}}$ then $p^{a} < 2n$?
My attempt:
$${{2n}\choose{n}} = \frac{(2n)!}{n!n!}$$
Let $a_1$ be the highest of power of $(2n)!$
Let $a_2$... | Hint: The highest power of a prime,$p$, that divides $n!$ is $\lfloor \frac{n}{p} \rfloor + \lfloor \frac{n}{p^2} \rfloor +\lfloor \frac{n}{p^3} \rfloor + \ldots \lfloor \frac{n}{p^k} \rfloor$. This is your $a_2$. Can you compare twice this with the expression for $2n$, your $a_1$?
| {
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"source": "stackexchange",
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Help complete a proof of Dirichlet on biquadratic character of 2? I am stuck proving the theorem that there exists $x$, $x^4 \equiv 2 \pmod p$ iff $p$ is of the form $A^2 + 64B^2$.
So far I have got this (and I am not sure if it's correct)
Let $p = a^2 + b^2$ be an odd prime,
*
*$\left(\frac{a}{p}\right) = \left(\f... | We can asume that 2 is a quadratic residue mod $p$ and so that $p \equiv 1 \pmod 8$ and this implies that if we pick $a$ odd and $b$ even then $b$ is a multiple of 4. We have to prove that $b$ is a multiple of 8.
First observe that as $x^2 \equiv -1 \pmod{p}$ and $a^2 + b^2 = p$ we have
$$ \left(\frac{a+b}{p}\rig... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Limit of $\lim\limits_{x \to\infty} 3\left(\sqrt{\strut x}\sqrt{\strut x-3}-x+2\right)$ I have to compute this limit:
$$\lim_{x \to\infty} 3(\sqrt{\strut x}\sqrt{\strut x-3}-x+2)$$
wolfram alpha says that answer is $\frac{3}{2}$, but I can't get why. Does anyone know how to get this limit?
| The two standard techniques work: multiply and divide by the conjugate, then divide both numerator and denominator by the highest power of $x$.
\begin{align*}
\lim_{x\to\infty}3\left(\sqrt{x}\sqrt{x-3} - x+2\right) &= 3\lim_{x\to\infty}\left(\sqrt{x^2-3x} - (x-2)\right)\\
&= 3\lim_{x\to\infty}\frac{(\sqrt{x^2-3x}-(x-2)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Inscribed kissing circles in an equilateral triangle
Triangle is equilateral (AB=BC=CA), I need to find AB and R.
Any hints?
I was trying to make another triangle by connecting centers of small circles but didn't found anything
| Let $a$ be the side of the triangle. If $A$ denotes the area and $P$ denotes the perimeter, then the radius of the incircle is given by $R = \frac{2A}{P} = \frac{2\sqrt{3} a^2/4}{3a} = \frac{\sqrt{3} a}{6}$
Let $x$ be the distance of the center of the smaller circle to the nearest vertex.
The altitude is $x + 8 + 2R = ... | {
"language": "en",
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Find limit of $\sqrt[n]{a^n-b^n}$ as $n\to\infty$, with the initial conditions: $a>b>0$ With the initial conditions: $a>b>0$;
I need to find $$\lim_{n\to\infty}\sqrt[n]{a^n-b^n}.$$
I tried to block the equation left and right in order to use the Squeeze (sandwich, two policemen and a drunk, choose your favourite) theor... | Here is a short solution based on standard inequalities.
Our first inequality is obvious since $b^n>0$
$$(1)\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad a^n-b^n\leq a^n.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad$$
Next we note that
$$a^n-b^n = a... | {
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$2$ equations with $4$ variables Given the equations,
$$v_1+v_2=a-1$$
$$v_1v_2=b$$
for what ranges of $a$ and $b$, can I be sure to find $0<v_1<1$ and $0<v_2<1$.
Also, for what ranges, can I be sure to find at least one $v_i$ such that $0<v_i<1$
| The solution is $v_1=\frac{1}{2}(-1+a-\sqrt{(a-1)^2-4b}), v_2=\frac{1}{2}(-1+a+\sqrt{(a-1)^2-4b})$ or the same with $v_1, v_2$ reversed. So we need $\frac{(a-1)^2}{4} \gt b$ to keep the square roots real. Then we need $1 \lt a-\sqrt{(a-1)^2-4b}$, which with $b \gt 0$ gives $a \gt 1$. We also need $a+\sqrt{(a-1)^2-4... | {
"language": "en",
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Multiplication in Permutation Groups Written in Cyclic Notation I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, if
$$
a=(1\,3\,5\,2),\quad b=(2\,5\,6),\quad c=(1\,6\,3\,4),
$$
then why does $ab=(1\,3\,5\,6)$ and $ac=(1\,6\,5\,2)(3\,4)$?
| You are thinking of the permutations as functions, so when you write "$ab$", you mean that you perform the permutation $b$ first, and the permutation $a$ second.
Here's one way to do it: write the disjoint cycle expressions for both $a$ and $b$, in the given order:
$$(1,3,5,2)(2,5,6)$$
Now, moving from right to left, s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/31763",
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"source": "stackexchange",
"question_score": "54",
"answer_count": 4,
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Positive integer solutions of $x^2+21y^2=z^4 $ Can one find all positive integer solutions of
$$x^2+21y^2=z^4 ?$$
I am not sure if this is possible. I just saw this problem and this problem came to my mind.
| This is an old post, but I believe a clarification to the other answer might be useful. The equation,
$$x^2+dy^2 = z^k\tag{1}$$
for $k = 2$ has a complete solution in terms of the single formula,
$$((p^2-dq^2)u)^2+d(2pqu)^2=((p^2+dq^2)u)^2\tag{2}$$
where $u$ is a scaling factor. However, the situation when $k>2$ is dif... | {
"language": "en",
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If $\alpha$ is an acute angle, show that $\int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$ If $\alpha$ is an acute angle, show that $\displaystyle \int_0^1 \frac{dx}{x^2+2x\cos{\alpha}+1} = \frac{\alpha}{2\sin{\alpha}}.$
My attempt:
Write $x^2+2x\cos{\alpha}+1 = (x+\cos{\alpha})^2+1-\cos^2{\a... | Or you can use this: $$\tan^{-1}{x} - \tan^{-1}{y} = \tan^{-1}\biggl(\frac{x-y}{1+xy}\biggr)$$
I am doing just the calculation part. We have
\begin{align*} \frac{1}{1+ \frac{\cos\alpha+\cos^{2}\alpha}{\sin^{2}\alpha}}\times \frac{1 + \cos\alpha}{\sin\alpha} - \frac{\cos\alpha}{\sin\alpha} &= \frac{1}{\sin\alpha} \time... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Divisibility of 9 and $(n-1)^3 + n^3 + (n+1)^3$ Question: Show that for all natural numbers $n$ which greater than or equal to 1, then 9 divides $(n-1)^3+n^3+(n+1)^3$.
Hence, $(n-1)^3+n^3+(n+1)^3 = 3n^3+6n$, then $9c = 3n^3+6n$, then $c=(n^3+2n)/3$.
Therefore $c$ should be integers, but I don't know how to do it at nex... | From your computations, it's enough to show that $n^3+2n=n(n^2+2)\equiv 0\pmod{3}$. If $n\equiv 0\pmod{3}$, you are done, else $n\equiv 1,2\pmod{3}$, in which case $n^2+2\equiv 0\pmod{3}$.
Reducing it to modular arithmetic may save you a lot of messy multiplication.
| {
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Problem: Sum of absolute values of polynomial roots Can you please give me some hints as to how I might approach this problem? Thanks!
Given the polynomial $f = 2X^3 - aX^2 - aX + 2, a \mathbb \in R$ and roots $x_1, x_2$ and $x_3,$ find $a$ such that $|x_1| + |x_2| + |x_3| = 3.$
Edit: We know $-1$ is one of the roots... | HINT
$|x_1| + |x_2| + |x_3| \geq 3 \sqrt[3]{|x_1||x_2||x_3|} = 3$
Hence, we have $|x_1| + |x_2| + |x_3| \geq 3$. Equality holds implies $|x_1| = |x_2| = |x_3| = 1$
We have $x_1 + x_2 + x_3$, $x_1x_2 + x_2x_3 + x_3x_2$ and $x_1 x_2 x_3$ to be real, and further $x=-1$ satisfies the equation.
Hence $f(x) = 2 \left( x+1 \... | {
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Minimal polynomial of $\frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$ I am struggling to find the minimal polynomial for $\displaystyle \frac{\sqrt{2}+\sqrt[3]{5}}{\sqrt{3}}$.
Does anyone have any suggestions?
Thanks,
Katie.
| One can compute the minimal polynomial using resultants or Grobner bases. But that is a bit overkill here since it can be done fairly straightforwardly by hand. Namely, let $\rm\ y = \sqrt{2}+\sqrt[3] 5\:.\ $ Then $\rm\: (y-\sqrt 2)^3 = 5\:,\:$ i.e. $\rm\:y^3 + 6\ y - 5 - (3\ y^2 + 2)\ \sqrt{2} = 0\:.\:$ Multiplying th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Power series of $\ln(x+\sqrt{1+x^2})$ without Taylor The answer is $$x-\frac{ 1}{2}\frac{ x^3}{3}+\frac{ 1\cdot 3}{2\cdot 4}\frac{ x^5}{5}-\frac{ 1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{ x^7}{7}+\cdots$$ But I can't see how. Unfortunately, "how" can't be using Taylor's formula, because that isn't introduced until the ne... | Do you know how to expand $1/\sqrt{1+x^2}$? See for instance this Wikipedia article.
Updated to add: You don't need calculus to derive this series: $$\frac{1}{\sqrt{1-y}} = 1+\frac{ 1}{2} y+\frac{ 1\cdot 3}{2\cdot 4} y^2+\frac{ 1\cdot 3\cdot 5}{2\cdot 4\cdot 6} y^3+\cdots$$ Just square the right-hand side, and you will... | {
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Solving trigonometric equation involving summation For $ 0 <\theta<\frac{\pi}{2}$ find the solution of
$$\sum\limits_{m=1}^{6}\csc\left(\theta+\frac{(m-1)\pi}{4}\right)\cdot\csc\left(\theta+\frac{m\pi}{4}\right)=4\sqrt{2}$$
I thought of solving this as the angles form an A.P , But the given sum does not
come under any ... | The left side of the equation can be rewritten as:
$$ \Delta = \sum_{1 3 5} \csc\left(\theta+\frac{m\pi}{4}\right) \cdot \left( \csc\left(\theta+\frac{(m-1)\pi}{4}\right) + \csc\left(\theta+\frac{(m+1)\pi}{4}\right) \right) $$
Now using the formulas
$$ \sin u +\sin v = 2\sin\left(\frac {u + v} 2\right) \cdot \cos \lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/46508",
"timestamp": "2023-03-29T00:00:00",
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Blockwise inversion case when $\textbf{D} - \textbf{C}\textbf{A}^{-1}\textbf{B}$ is singular What means in blockwise matrix inversion when $\textbf{D} - \textbf{C}\textbf{A}^{-1}\textbf{B}$ is singular but $\textbf{A}$ is not? is that necessary and sufficient for the whole composed matrix be singular as well? are there... | It cannot happen. You are starting with a block matrix of the form
$$\mathbf{M}=\left(\begin{array}{cc}
\mathbf{A} & \mathbf{B}\\
\mathbf{C} & \mathbf{D}
\end{array}\right),$$
which we are assuming is invertible; that is, $\det(\mathbf{M})\neq 0$.
But if $\mathbf{A}$ is invertible, then we also have
$$\det(\mathbf{M... | {
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"timestamp": "2023-03-29T00:00:00",
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Expansion concerning the binomial theorem The question goes:
Expand $(1-2x)^{1/2}-(1-3x)^{2/3}$ as far as the 4th term.
Ans: $x + x^2/2 + 5x^3/6 + 41x^4/24$
How should I do it?
| Substituting $r=1/2$ and $y=-2x$ in the binomial series
$$
(1 + y)^r = 1 + \frac{r}{{1!}}y + \frac{{r(r - 1)}}{{2!}}y^2 + \frac{{r(r - 1)(r - 2)}}{{3!}}y^3 + \frac{{r(r - 1)(r - 2)(r - 3)}}{{4!}}y^4 + \cdots
$$
gives
$$
(1-2x)^{1/2} = 1 - x - x^2/2 -x^3/2 - (5/8)x^4 + \cdots,
$$
while substituting $r=2/3$ and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove that $\lim\limits_{(x,y) \to (0,0)} \frac{\left | x \right |^{\frac{3}{2}}y^{2}}{x^{4} + y^{2}} \rightarrow 0$ How can I prove that
$$\lim_{(x,y)\to (0,0)} \frac{\left | x \right |^{\frac{3}{2}}y^{2}}{x^{4} + y^{2}} \rightarrow 0\;?$$
Thanks!
| For $y \neq 0$,
$$
0 \le \frac{{|x|^{3/2} y^2 }}{{x^4 + y^2 }} = \frac{{|x|^{3/2} }}{{x^4 /y^2 + 1}} \le \frac{{|x|^{3/2} }}{1} = |x|^{3/2} \to 0.
$$
EDIT: In retrospect, simply note that $0 \le \frac{{y^2 }}{{x^4 + y^2 }} \le 1$, for $(x,y) \neq (0,0)$, to conclude that $\frac{{|x|^{3/2} y^2 }}{{x^4 + y^2 }} \t... | {
"language": "en",
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Limit of this series: $\lim_{n\to\infty} \sum^n_{k=0} \frac{k+1}{3^k}$? Given a series, how does one calculate that limit below? I noticed the numerator is an arithmetic progression and the denominator is a geometric progression — if that's of any relevance —, but I still don't know how to solve it.
$$\lim_{n\to\infty}... | divide that formular like this.
$$\sum_{k=1}^{\infty}\left (\frac{k}{3^k}+\frac{1}{3^k}\right )$$
then $$\sum_{k=1}^{\infty}\frac{k}{3^k}+\frac{\frac{1}{3}}{1-\frac{1}{3}}=\sum_{k=1}^{\infty}\frac{k}{3^k}+\frac{1}{2}$$
power series
$$ \sum_{k=1}^{\infty}\frac{k}{3^k}$$
let $$ \sum_{k=1}^{\infty}\frac{k}{3^k}=S$$
Then,
... | {
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A "fast" way to ,find the maximum value of $(x^2) \times (y^3)$, if $3x+4y=12$ for $x,y \ge 0$ If $3x+4y=12$ $\forall x,y \ge 0$, the maximum value of $(x^2) \times (y^3)$ is
*
*$6 \times (6/5)^5$
*$3 \times (6/5)^5$
*$ (6/5)^5 $
*$7 \times (6/5)^5$
How to approach this problem? I thought of using the approac... | Lagrange multipliers are pretty fast, and don't require seeing any tricks:
$$2xy^3 = 3\lambda$$
$$3x^2y^2 = 4\lambda$$
Divide the two equations, and you get ${\displaystyle {2 \over 3} {y \over x} = {3 \over 4}}$ or ${\displaystyle y = {9 \over 8} x}$. Putting back into the equation you get
$$3x + {9 \over 2} x = 12$$
... | {
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Precision of operations on approximations If $ x $ and $ y $ have $ n $ significant places, how many significant places do $ x + y $, $ x - y $, $ x \times y $, $ x / y $, $ \sqrt{x} $ have?
I want to evaluate expressions like $ \frac{ \sqrt{ \left( a - b \right) + c } - \sqrt{ c } }{ a - b } $ to $ n $ significant pla... | $ \left[ a , b \right] + \left[ c , d \right] = \left[ a + c , b + d \right] $
$ \left[ a , b \right] - \left[ c , d \right] = \left[ a - d , b - c \right] $
$ \left[ a , b \right] \times \left[ c , d \right] = \left[ \min \left( a \times c , a \times d , b \times c , b \times d \right) , \max \left( a \times c , a \t... | {
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Exponential Diophantine perfect square problem I need to find all positive integer solutions $(a,b,c)$ to $1+4^a+4^b=c^2$. I am certain that the only solutions are $(t,2t-1, 2^{2t-1}+1) $ and $ (2t-1, t, 2^{2t-1}+1) $ , $t\in \mathbb{N}$ but I am having some trouble confirming this. Could someone help me please? Thank... | By symmetry it is enough to deal with the case $1 \le a \le b$.
Rewrite the equation as
$$4^a(1+4^{b-a})=c^2-1=(c-1)(c+1).$$
Suppose first that $b=a$. Then
$4^a(1+4^{a-b})=2^{2a+1}$. But $(c-1)(c+1)$ is a power of $2$ only if $c=3$. Thus $a=b=1$. From now on we can assume that $b>a$.
Since $a \ge 1$, $c$ must be o... | {
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"timestamp": "2023-03-29T00:00:00",
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Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$ Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that:
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}... | We can use the Vasc's EV Method.
See here: https://www.emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf
Indeed, let $a+b+c+d+e=constant.$
Thus, by corollary 1.9, case 1(b) ($p=0$,$q=-1$)
the expression $\sum\limits_{cyc}\frac{1}{a}$ gets a minimal value for equality case of four variables.
Id est, it remains to p... | {
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"answer_id": 1
} |
Finding limit of a quotient I can't find this for some reason. I know I asked about 6 of these before, and I was able to finish my homework but now I went back to review and I can't do a single one of these problems on my own. Even the ones I did figure out on my own. I spent probably a total of 14 hours on the homewor... | For $x=-2$,
$$\begin{eqnarray*}
x+2 &=&0 \\
x^{3}+8 &=&0,
\end{eqnarray*}$$
which means $x=-2$ is a root of both equations. Hence $x^{3}+8$ may be
factored$^1$ as
$$x^{3}+8=(x-(-2))Q(x)=(x+2)Q(x).\qquad(*)$$
To compute $Q(x)$
*
*you may perform the long division $(x^{3}+8):(x+2)$ (or apply the so called Riffini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
} |
Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction I recently proved that
$$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$
using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also l... | For every $k\in\mathbb{N}$
$$(k+1)^4=k^4+4k^3++6k^2+4k+1$$
therefore
$$\sum_{k=1}^n(k+1)^4=\sum_{k=1}^nk^4+4\sum_{k=1}^nk^3+6\sum_{k=1}^nk^2+4\sum_{k=1}^nk+\sum_{k=1}^n1$$
which is equivalent to
$$\sum_{k=1}^nk^4+(n+1)^4-1=\sum_{k=1}^nk^4+4\sum_{k=1}^nk^3+6\sum_{k=1}^nk^2+2n(n+1)+n$$
After simplifications we obtain
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/61482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "184",
"answer_count": 28,
"answer_id": 7
} |
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
|
This picture shows that $$1^2=1^3\\(1+2)^2=1^3+2^3\\(1+2+3)^2=1^3+2^3+3^3\\(1+2+3+4)^2=1^3+2^3+3^3+4^3\\$$ this is handmade of mine
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/62171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "67",
"answer_count": 16,
"answer_id": 11
} |
Asymptotic behaviour of sums of consecutive powers Let $S_k(n)$, for $k = 0, 1, 2, \ldots$, be defined as follows
$$S_k(n) = \sum_{i=1}^n \ i^k$$
For fixed (small) $k$, you can determine a nice formula in terms of $n$ for this, which you can then prove using e.g. induction. For small $k$ we for example get
$$\begin{ali... | This is a classic application of the Euler-Maclaurin formula for approximating a sum by an integral. Euler-Maclaurin says
$$\sum_{i=0}^n f(i) = \int_0^n f(x) dx + \frac{f(n)+f(0)}{2} + \sum_{i=1}^{\infty} \frac{B_{2i}}{(2i)!} \left(f^{(2i-1)}(n)-f^{(2i-1)}(0)\right),$$
where $B_i$ is the $i$th Bernoulli number.
If we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/63986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 0
} |
Solving the equation $x + \sqrt{2x+1} = 7$ I can't solve
$x + \sqrt{2x+1} = 7$.
Well, I know the answer is 4, but that is from just reasoning it out. I can't algebraically solve it.
Thus, a step by step is what I really need.
Thanks in advance!!
| If $x + \sqrt{2x+1} = 7$, multiply both sides by $x - \sqrt{2x+1}$
to get $x^2 - (2x+1) = 7(x - \sqrt{2x+1})$, or
$$\sqrt{2x+1} = x - (x^2 - (2x+1))/7 = (-x^2 + 9x+1)/7.$$
Substituting in the original equation,
$7 = x + (-x^2 + 9x+1)/7 = (-x^2 + 16x+1)/7$
or $x^2 - 16x + 48 = 0$.
Solving this, we get $x = 4$ and $x = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/64811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How to show that $91$ is a pseudoprime to the base $3$? The given problem:
Use Lemma 2.3.3 together with Fermat's little theorem to show that 91
is a pseudoprime to the base 3.
Lemma 2.3.3.
Let $m_1 \dots m_r \in $ N. If $a \equiv b \pmod {m_i}$, $\forall i =1, \dots, r$, then $a \equiv b \pmod {{\rm lcm}(m_1, \dot... | Powers of $3$ cycle as follows
$$
\begin{array}{|c|c|c|}
n&3^n&3^n \mod 91\\
\hline\\
1 & 3 & 3\\
2 & 9 & 9\\
3 & 27 & 27\\
4 & 81 & 81\\
5 & 243 & 61\\
6 & 729 & 1\\
\hline
\end{array}
$$
Therefore, since $90 = 6 \times 15$
$$
\begin{align*}
3^{90} &\equiv 1 \hspace{4pt} (\mod 91)\\
\Rightarrow 3^{91} &\equiv 3 \hspa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/67924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A coin is flipped when dice hits 6 (conditional probability) Prof gave us homework on conditional probability that is due on the day of the lecture on conditional probability. Yeah, this has been a bad week and I've no idea what I'm doing.
Q: 3 dice are rolled, then, a coin is flipped as many times as the number 6 is ... | a) The computation takes a while. It may be useful to draw a tree in order not to lose track of the possibilities. Initially, we toss $3$ dice. We get $0$, $1$, $2$, or $3$ $6$'s. Then, depending on the outcome, we toss a certain number of coins. So from the "start" position, there are $4$ branches, corresponding ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/69902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
probablity of random pick up three points inside a regular triangle which form a triangle and contain the center what is the probablity of random pick up three points inside a regular triangle
which form a triangle and contain the center of the regualr triangle
the three points are randomly picked within the regular ... | Let $R$ be a convex region of the plane with unit area, and choose $n\ge 3$ points at random from the region. We will derive a general expression for the probability $P_{n}$ that the convex hull of these points contains a particular point $\tau \in R$ (which we will take as the origin of our coordinate system). We can... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/72977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 5,
"answer_id": 3
} |
How many ordered pairs of positive integers
For a prime integer p, how many ordered pairs of positive integers
(a, b) are there that satisfy $$\frac{1}{a} + \frac{1}{b} =\frac{1}{p}$$
For example, for p = 5, $$\frac{1}{6} + \frac{1}{30}$$ and
$$\frac{1}{30} + \frac{1}{6}$$ are two different ways of getting
$\fra... | $1/a+1/b=1/p$, $bp+ap=ab$, $(a+b)p=ab$, so $p$ divides $a$ or $p$ divides $b$ (or both). Let's say $p$ divides $a$, so $a=cp$ for some $c$. Now $cp+b=bc$, $b=bc-pc=(b-p)c$, so $c$ divides $b$. Let $b=cd$. Then $d=cd-p$, so $p=cd-d=(c-1)d$. So either $c-1=p$ and $d=1$, or $c-1=1$ and $d=p$. Now work your way back up to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/73496",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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$n! \leq \left( \frac{n+1}{2} \right)^n$ via induction I have to show $n! \leq \left( \frac{n+1}{2} \right)^n$ via induction.
This is where I am stuck:
$$\left( \frac{n+2}{2} \right)^{n+1}
\geq \dots \geq
=2 \left( \frac{n+1}{2} \right)^{n+1}
= \left( \frac{n+1}{2} \right)^n(n+1)
\geq n!(n+1)
= (n+1)! $$
I appro... | Hint:
$$
(n+1)! = (n+1) n! \leq (n+1) \left( \frac{n+1}{2} \right)^n = 2 \left( \frac{n+1}{2} \right)^{n+1}.
$$
You can check that $2 \left( \frac{n+1}{2} \right)^{n+1} \leq \left( \frac{n+2}{2} \right)^{n+1}$, by proving that
$$
2 \leq \left( \frac{n+2}{n+1} \right)^{n+1}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/76130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 0
} |
Integrate $\int\frac{1}{x^6} \sqrt{(1-x^2)^3} ~ dx$ How to integrate the following?
$$\int\frac{\sqrt{(1-x^2)^3}}{x^6} \;dx .$$
| Hint: Do the substitution $x = \sin(\alpha)$, $\alpha \in [-\frac{\pi}{2},\frac{\pi}{2}]$. Then you get
$$\int \frac{\cos^4(\alpha)}{\sin^6(\alpha)} d\alpha.$$
Now consider the following identities:
$$D\left(\frac{\cos^3(\alpha)}{\sin^5(\alpha)}\right) = \frac{-3\cos^2(\alpha)\sin^6(\alpha) - 5\sin^4(\alpha)\cos^4(\alp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Solving $(2y-4)(2y+1) = (2y-2)^2$ I'm getting different answer from answer key.
Solving $$(2y-4)(2y+1) = (2y-2)^2$$
FOIL left side $$4y^2+2y-8y-4 = (2y-2)^2$$
Right side $$4y^2+2y-8y-4 = 4y^2+4 $$
Subtract $4y^2$ from both sides
$$2y-8y-4 = 4 $$
Combine $y$
$$6y-4 = 4$$
add 4 to both sides
$$6y = 8$$
But the answer key... | The error is in the "Right Side" step.
You essentially wrote
$$(2y-2)^2 = 4y^2 + 4.$$
That's incorrect.
Remember: $(a-b)^2 = a^2 - 2ab + b^2$. So
$$(2y-2)^2 = 4y^2 - 8y + 4.$$
The third displayed equation should thus be
$$4y^2 +2y - 8y - 4 = 4y^2 -8y + 4.$$
You will find that this leads to $2y = 8$, from which you get ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
quotient rule difficulties I'm trying to use the quotient rule to differentiate $\frac{r}{\sqrt{r^2+1}}$ but I'm getting the wrong answer. So far I have
$$\begin{align*}
\frac {d}{dr} \frac{r}{\sqrt{r^2+1}} &= \frac {\sqrt{r^2+1} \frac {d}{dr} r - r \frac {d}{dr} \sqrt{r^2+1}} {(\sqrt{r^2+1})^2} \\\\\\\\
&= \frac {\... | There's nothing wrong with your application of the quotient rule. You just need to simplify your answer further:
$$
\begin{eqnarray*}
(r^2+1)^{-1/2} - r^2 (r^2+1)^{-3/2}
&=& (r^2+1) \cdot (r^2+1)^{-3/2} - r^2 \cdot (r^2+1)^{-3/2}
\\ &=& (r^2+1)^{-3/2} \cdot \left((r^2+1) - r^2 \right)
\\ &=& (r^2+1)^{-3/2} \cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/77724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
If $\gcd(a,b)=1$ , and $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$? How to prove that:
$\gcd(a,b)=1 \Rightarrow \gcd(2^{a}+1,2^{b}+1)=1$ ,where $a$ is even and $b$ is odd natural number
For example:
$\gcd(2^8+1,2^{13}+1)=1 , \gcd(2^{64}+1,2^{73}+1)=1$
I know that Knuth showed that:
$\gcd(2^{a}-1,2^{b}-1... | This is a minor tweak of my answer to an earlier post by user952949. That one asked for a proof that if $a$ and $b$ are relatively prime and odd, then $\gcd(2^a+1,2^b+1)=3$.
A very useful fact: If $a$ and $b$ are relatively prime, there exist integers $x$ and $y$ such that $ax+by=1$.
We can arrange for $x$ to be $\g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Proving that $f(n)$ is an integer using mathematical induction I want to prove that
$$\frac{n^3}{3}+\frac{n^5}{5}+\frac{7 n}{15}$$
is an integer for every integer $n \geq 1$.
I define P(n) to be:
$$\frac{n^3}{3}+\frac{n^5}{5}+\frac{7 n}{15}$$ is an integer.
For my basis step, P(1) is true because
$$\frac{1^3}{3}+\frac... | Why do you think that $P(k) = 15m$ for some integer $m$ if it does not hold for, say $k=1$? If you assume that $P(k)$ is integer then the strategy is to show that
$$
P(k+1) - P(k) \in\mathbb Z
$$
and let us do it:
$$
P(k+1) - P(k) = \frac{1}{5}((n+1)^5-n^5)+\frac13((n+1)^3 - n^3)+\frac7{15} =
$$
$$
= \frac15(5n^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Determine $z^n+z^{-n}$ if $z+\frac{1}{z}=-2\cos{x}$ Determine $z^n+z^{-n}$ if $z+\frac{1}{z}=-2\cos{x}$ with $z \in \mathbb{C}$.
| Since $z + \frac{1}{z} = - 2 \cos(x)$ is equivalent to $z^2 + 2 z \cos(x) + 1 = 0$, it is solved by $z_{1,2} = -\cos(x) \pm i \sqrt{1-\cos^2(x)}$. Since $1-\cos^2(x) = \sin^2(x)$, these also solve the equation $\tilde{z}_{1,2} = -\cos(x) \mp i \sin(x) = -\exp(\pm i x)$.
Now to find $z^n+z^{-n}$ for $z$ being the soluti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/78605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
integrate square of $\arctan x$. Tricky $$\int \left(\frac{\tan^{-1}x}{x-\tan^{-1}x}\right)^{2}dx$$
I ran across an integral I am having a time solving.
The solution merely works out to $\displaystyle\frac{1+x\tan^{-1}x}{\tan^{-1}x-x}$, but for the life of me I can not find a suitable method to tackle it.
Does anyone ... | With $x = \tan(u)$, $\mathrm{d}x = \frac{1}{\cos^2(u)}\mathrm{d} u$, thus
$$
\int \left(\frac{\tan^{-1}x}{x-\tan^{-1}x}\right)^{2} \mathrm{d}x =
\int \left( \frac{u}{\tan(u) - u} \cdot \frac{1}{\cos(u)} \right)^2 \mathrm{d} u =
\int \left( \frac{u}{\sin(u) - u \cdot \cos(u)} \right)^2 \mathrm{d} u
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 2,
"answer_id": 0
} |
chances of getting three of one kind and four of another out of seven dice There are several questions similar to this one but after reading those,
I am still very confused.
I also did a similar problem of this one and I think I got it, but then I got stuck again.
So if four dice are rolled, the chance of getting thre... | Think of filling in 7 slots; in each you have the value of a die roll.
There are $6\cdot 5$ ways to choose the values for the three and, different valued, four of a kind (for example,
the three of a kind is three '2's and the four of a kind is four '5's).
There are $7\choose 4$ ways to select ''slots'' in which to plac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/79637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Modular exponentiation by hand ($a^b\bmod c$) How do I efficiently compute $a^b\bmod c$:
*
*When $b$ is huge, for instance $5^{844325}\bmod 21$?
*When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$?
*When $(a,c)\ne1$, for instance $6^{103... | Here are two examples of the square and multiply method for $5^{69} \bmod 101$:
$$ \begin{matrix}
5^{69} &\equiv& 5 &\cdot &(5^{34})^2 &\equiv & 37
\\ 5^{34} &\equiv& &&(5^{17})^2 &\equiv& 88 &(\equiv -13)
\\ 5^{17} &\equiv& 5 &\cdot &(5^8)^2 &\equiv& 54
\\ 5^{8} &\equiv& &&(5^4)^2 &\equiv& 58
\\ 5^{4} &\equiv& &&(5^2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "128",
"answer_count": 11,
"answer_id": 5
} |
Which is the "fastest" way to compute $\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $?
I am looking for the "fastest" paper-pencil approach to compute $$\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $$
This is a quantitative aptitude problem and the correct/required answer is $3.75$
In addition, I am also interested... | The sum $S= \sum \limits_{i=1}^n \frac{1}{2^i}=1-\frac{1}{2^n}$ is geometric, thus easy to calculate.
Here is a simple elementary way of calculating
$$T=\sum_{i=1}^n \frac{i}{2^i} \,.$$:
$$T=\sum_{i=1}^n \frac{i}{2^i} =\frac{1}{2}+ \sum_{i=2}^n \frac{i}{2^i} =\frac{1}{2}-\frac{n+1}{2^{n+1}}+ \sum_{i=2}^{n+1} \frac{i}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/81362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$ $5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$
Work out the values of $\frac{1}{x+y}$
| My solution is quite elementary.
So we consider the equations $$5^x + 2^y = \frac {7}{10} = 2^x +5^y $$
Note that $5^x + 2^y = \frac {7}{10} ( eq.1 ) $ and $\frac {7}{10} = 2^x +5^y (eq.2) $ are inverse functions.
By inspection, we can see that $(-1, -1 )$ is a solution. Now, If we show that this is the only soluti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/83881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 6,
"answer_id": 3
} |
Existence of the limit of a sequence? I solved this limit problem by following this way, but I'm not exactly sure about ....
can anyone help me and tell me if it is correct?
the problem is:
Let $k>1$. If it exists, calculate the limit of the sequence $(x_n)$,
$$x_n := \Biggl(k \sin (\frac{1}{n^2}) + \frac{1}{k}\cos ... | The limit is zero. You can argue as follows. If $n$ is large enough that $1/n^2 \leq \pi/2$, then $\sin(1/n^2)\leq 1/n^2$. Then we have that
$$|x_n| \leq \left(\frac{k}{n^2} + \frac{1}{k}\right)^n.$$
Then choose $n$ large enough so that $\frac{k}{n^2} + \frac{1}{k} \leq 1-\delta$ for some positive $\delta<1$. This ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/84571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
random variable transformation I'm having trouble with the following random variable transformation:
$Y = X^2 + X$
I am looking for the pdf of Y. I tried the following method:
$p_Y(y) = \int_{X} p_{Y|X=x}(y)\cdot p_{X}(x)dx$
and we know that $(Y|X=x) \sim (x^2+x) \Rightarrow p_{Y|X=x} = \delta_{x^2+x}(y)$
thus: $p_Y(y)... | Let $Y = X^2 + X = \left( X+\frac{1}{2} \right)^2 - \frac{1}{4}$. Then
$$
F_Y(y) = \mathbb{P}(Y \le y) = \mathbb{P}\left( \left( X+\frac{1}{2} \right)^2
\le y + \frac{1}{4} \right)
$$
Assume, additionally, that $y+\frac{1}{4} > 0$. Then
$$ \begin{eqnarray}
F_Y(y) &=& \mathbb{P}\left( -\sqrt{y + \frac{1}{4}} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/85037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Probability that a family with $n$ children has exactly $k$ boys Let the probability $p_n$ that a family has exactly $n$ children be $\alpha p^n$ when $n\geq1$, and $$p_0=1-\alpha p(1+p+p^2+\cdots).$$ Suppose that all the sex distributions have the same probability. Show that for $k\geq1$ the probability that a family ... | Extended hint: We sketch an argument that uses only basic notions.
Note that the probability $b_k$ of $k$ boys is, by a conditional probability argument, given by
$$b_k=\sum_{n=1}^\infty \alpha p^n \binom{n}{k} \left(\frac{1}{2}\right)^{k}\left(\frac{1}{2}\right)^{n-k}.$$
This simplifies to
$$b_k=\sum_{n=1}^\infty \a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/85733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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How do you divide a polynomial by a binomial of the form $ax^2+b$, where $a$ and $b$ are greater than one? I came across a question that asked me to divide $-2x^3+4x^2-3x+5$ by $4x^2+5$. Can anyone help me?
| I will try this way:
Since you are dividing a 3rd degree polynomial by a 2nd degree polynomial, WLOG, we may assume
$$-2x^3+4x^2-3x+5=(4x^2+5)(ax+b)+cx+d\quad(1)$$
Now, comparing the coefficients of $x^3$ and $x^2$ readily give $a=-\frac{1}{2}$ and $b=1$. Comparing coefficients of $x$, we have $5a+c=-3\Rightarrow c=-\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
How do you factor $x^3-3x^2+3x-1$? $$x^3-3x^2+3x-1?$$
I know this may seem trivial, but I, for the life of me, I cannot figure out how to factor this polynomial, I know that the root is $$(x-1)^3=0$$ because of wolframalpha, but I don't know how to get there. any help would be greatly appreciated. and also if you have ... | $$
\begin{align*}
x^3-3x^2+3x-1 &=x^3-x^2-2x^2+3x-1
\\ &=x^2(x-1)-2x^2+2x+x-1
\\ &=x^2(x-1)-2x(x-1)+1(x-1)
\\ &=(x-1)(x^2-2x+1)
\\ &=(x-1)(x-1)^2
\\ &=(x-1)^3
\end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/86352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 0
} |
Finding complex solutions of an equation How does one solve this equation. I would like to see the solution of this problem in steps.
$z\cdot\bar{z}=\left|3\cdot z \right|$
EDIT: Is it possible to solve this by converting to the form $z=a+b\cdot i$
What about the solution of this equation.
$z\cdot\bar{z}-z^{2}=1-i$
EDI... | First, note that for any complex number $z=a+bi$, we have
$$z\cdot \bar{z}=(a+bi)\cdot(a-bi)=a^2+abi-abi+b^2(i)(-i)=a^2+b^2=\left(\sqrt{a^2+b^2}\right)^2=|z|^2.$$
Now note that for any complex number $z=a+bi$ and real number $t$, we have
$$|t\cdot z|=|t(a+bi)|=|(ta)+(tb)i|=\sqrt{(ta)^2+(tb)^2}=\sqrt{(t^2)(a^2+b^2)}=$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/86771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to calculate $\sqrt{\frac{-3}{4} - i}$
Possible Duplicate:
How do I get the square root of a complex number?
I know that the answer to $\sqrt{\dfrac{-3}{4} - i}$ is $\dfrac12 - i$. But how do I calculate it mathematically if I don't have access to a calculator?
| For $a \ne 0$, there are two numbers $z$ such that $z^2=a$.
We look at the given example, using only basic tools. We want to solve the equation
$$z^2=-\frac{3}{4}-i.$$
Because of a general discomfort with negative numbers, we look at the equivalent equation
$$z^2=-\frac{1}{4}(3+4i).$$
In order to deal with simpler numb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Conditions for intersection of parabolas? What are the conditions for the existence of real solutions for the following equations:
$$\begin{align}
x^2&=a\cdot y+b\\
y^2&=c\cdot x+d\end{align}$$
where $a,b,c,d $ are real numbers.
These represent two parabolas; how might we find out the conditions for the existence of ... | Assume that $(x,y)$ is a point common to both parabolas. If we add the two equations together and complete the square we get the circle equation
\begin{equation*}
\left( x - (c/2) \right)^2 + \left(y - (a/2) \right)^2 = (a/2)^2 + (c/2)^2 + b + d
\end{equation*}
So one condition which is necessary for a solution is tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/92689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Find $ n\geq1 $ such that 7 divides $n^n-3$ Find $ n\geq1 $ such that 7 divides $n^n-3$.
Here is what I found:
$ n\equiv 0 \mod7, n^n\equiv 0 \mod7,n^n-3\equiv -3 \mod7$ no solution.
$ n\equiv 1 \mod7, n^n\equiv 1 \mod7,n^n-3\equiv -2 \mod7 $ no solution.
$ n\equiv 2 \mod7, n^n\equiv 2^n \mod7, n^n-3\equiv 2^n-3 \mod7$... | $$
5^5-3=2\cdot 7\cdot 233
$$
Other solutions are
$$n=31, 47, 73, 89, 115, 131, 157, 173, 199, 215, 241, 257, 283, 299, 325$$
All of them are odd and congruent to $3$ or $5$ modulo $7$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/93165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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Functional inverse of $\sin\theta\sqrt{\tan\theta}$ What is the functional inverse of $f(\theta) = \sin\theta\sqrt{\tan\theta}$? Or, equivalently, what is the inverse of
$$f(\theta)=\sin^2\,\theta\tan\,\theta=\frac{\sin^3\,\theta}{\cos\,\theta}$$
It comes from a physics setup involving two equivalently massed and char... | I will assume you are interested in finding $\theta = f^{-1}(x)$ for $x \geq 0$ with the range $0 \leq \theta < \frac{\pi}{2}$.
$$
x^2 = \left(f(\theta)\right)^2 = \sin^2(\theta) \tan(\theta) = \frac{\tan^3(\theta)}{1+\tan^2(\theta)}
$$
Hence $\theta = \arctan(y(x))$, where $y$ is the positive root of $y^3 = x^2 (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/93509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Something interesting that I found about some numbers - and would like to see if it's known Well I am quite sure it's known (I mean number theory exists thousands of years), warning beforehand, it may look like numerology, but I try not to go to mysticism.
So I was in a bus, and from boredom I started just adding numbe... | A repetition of this sort was bound to happen, and it always happens even under more general circumstances.
First, as others have pointed out, the sum of the base-10 digits of a number $N$ is congruent to $N$ modulo $9$. The reason for this is that $10\equiv 1 \bmod 9$, and so
$$\begin{align*} N &= a_t \cdot 10^t + a_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/94997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 4,
"answer_id": 2
} |
Minimal polynomial of $1 + 2^{1/3} + 4^{1/3}$ I am attempting to compute the minimal polynomial of $1 + 2^{1/3} + 4^{1/3}$ over $\mathbb Q$. So far, my reasoning is as follows:
The Galois conjugates of $2^{1/3}$ are $2^{1/3} e^{2\pi i/3}$ and $2^{1/3} e^{4\pi i /3}$. We have $4^{1/3} = 2^{2/3}$, so the image of $4^{1/... | By expanding and using the relation $1+e^{2\pi i/3}+e^{4\pi i/3}=0$ heavily I got that
$$
(x-a)(x-b)(x-c)=x^3-3x^2-3x-1.
$$
Looks like rational coefficients to me.
Another way of seeing this is to compute
$$
(a-1)^3=(2^{1/3}+4^{1/3})^3=2+3\cdot 2^{4/3}+3\cdot 2^{5/3}+4=6+6(2^{1/3}+4^{1/3})=6+6(a-1)=6a.
$$
Hence
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/95918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
How to prove $\lim\limits_{n \to \infty} (1+\frac1n)^n = e$?
How to prove the following limit?
$$\lim_{n \to \infty} (1+1/n)^n = e$$
I can only observe that the limit should be a very large number!
Thanks.
| Actually, the way things work out in mathematics usually is that we only prove that $x_n = (1+1/n)^n$ is a convergent sequence, and we define its limit to be $e$. Some people use other definitions for $e$ and show that it is equivalent to this definition, but there are many ways to do this that are logically equivalent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/96606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Approximate $\int_{0}^{\infty} \frac{\text{d} x}{1 + x^4}$ Now, I have been given this integral. And need to approximate it. My first idea was to use a Taylor series, but this series explodes, as x reaches infinity.
Does anyone know how to approximate improper integrals, (and this one in particular)?
I know I can use ... | One way is to split integration range at $x=1$ and use geometric series approximation:
$$\begin{eqnarray}
\int_0^\infty \frac{\mathrm{d} x}{1+x^4} &=& \int_0^1 \frac{\mathrm{d} x}{1+x^4} +\int_1^\infty \frac{\mathrm{d} x}{1+x^4} \stackrel{x -> 1/x \text{ in second}}{=}
\\ &=& \int_0^1 \frac{\mathrm{d} x}{1+x^4}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/99265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
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Solving $y'' - xy'+(3x-2)y=0$ using power series I am trying to solve this equation using the series $$\sum_0^\infty a_nx^n$$
$$y'' - xy'+(3x-2)y=0$$
How to do that? I mean that I can replace the variables using the series but then I cannot add this thing cause the limits of the sums are not the same. Maybe I am doing... | Let
$$
y=\sum\limits_{n=0}^\infty a_n x^n
$$
then by a straightforward computation we get
$$
y''-xy'+(3x-2)y=\sum\limits_{n=2}^\infty n(n-1)a_n x^{n-2}-x\sum\limits_{n=1}^\infty n a_n x^{n-1}+3x\sum\limits_{n=0}^\infty a_n x^n-2\sum\limits_{n=0}^\infty a_n x^n=
$$
$$
\sum\limits_{n=0}^\infty (n+2)(n+1)a_{n+2} x^n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/102549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Groups of units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ On page 230 of Dummit and Foote's Abstract Algebra, they say: the units of $\mathbb{Z}\left[\frac{1+\sqrt{-3}}{2}\right]$ are determined by the integers $a,b$ with $a^2+ab+b^2=\pm1$ i.e. with $(2a+b)^2+3b^2=4$, from which is is easy to see the group of u... | The reason you may want to change it from $a^2+ab+b^2=\pm 1$ to $(2a+b)^2+3b^2 = \pm 4$ is because the latter is a sum of squares, so this immediately cuts down on the possibilities: for one thing, you can tell that the answer must be $4$ and not $-4$ (sum of squares), that you must have $|2a+b|\leq 2$ and $3b^2\leq 4$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/105097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
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Rigorous proof of an infinite product. I'll give a proof of the following expansion:
$$\frac{\sin x}{x} = \prod_{i=1}^{\infty} \cos \frac{x}{2^i}$$
$${\sin x} = 2 \cos \frac{x}{2}\sin \frac{x}{2}$$
$${\sin x} = 2^2 \cos \frac{x}{2}\cos \frac{x}{4}\sin \frac{x}{4}$$
$$ {\sin x} = 2^3 \cos \frac{x}{2} \cos \frac{x}{4}... | Your expression
$$ \frac{\sin x}{x} = \frac{\sin \frac{x}{2^k}}{ \frac{x}{2^k}} \prod_{i=1}^{k} \cos \frac{x}{2^i} $$
is correct. Maybe we should separate out the very special case $x=0$, and from then on assume that $x\ne 0$. For $x=0$, $\frac{\sin x}{x}$ is formally undefined, but it is natural to set it equal to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/107144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ I have this exponential equation that I don't know how to solve:
$3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ with $x \in \mathbb{R}$
I tried to factor out a term, but it does not help. Also, I noticed that:
$2 \cdot 9^... | $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0 \Rightarrow 12 \cdot 2^{2x} - 35 \cdot 2^x \cdot 3^x + 18 \cdot 3^{2x} = 0 \Rightarrow$
$\Rightarrow 12 \cdot \left(\frac{2}{3}\right)^{2x}-35 \cdot \left(\frac{2}{3}\right)^{x}+18=0 $
Now make substitution : $\left(\frac{2}{3}\right)^{x} = t$ , and solve quadratic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/108447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 2
} |
Derive a formula to find the number of trailing zeroes in $n!$
Possible Duplicate:
How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?
I know that I have to find the number of factors of $5$'s, $25$'s, $125$'s etc. in order to do this. But how can you derive such a formul... | The number of trailing zeroes in $n!$ is the exponent of $5$ in the prime factorization of $n!$, which by the de Polignac's formula$^1$ is given by
$$e_5(n!)=\sum_{i= 1}^{\left\lfloor \log n/\log 5\right\rfloor} \left \lfloor \frac{n}{5^i} \right \rfloor.\tag{1}$$
Added. By the same de Polignac's formula the exponent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/111385",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 1
} |
Finding domain of $\sqrt{ \frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} }$ How can I find the domain of:
$$\sqrt{ \frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} }$$
I think the hard part will be to find:
$$\frac{(x^2-1)(x^2-3)(x^2-5)}{(x^2-2)(x^2-4)(x^2-6)} \ge 0$$
So far I have: not sure how to preceed:
$... | +1 for your handwriting :-)
*
*$\frac{f(x)}{g(x)} \ge 0$ is not so different from $f(x)\times g(x) \ge 0$ (except for zeros of $g$.)
*With $(x^2-1) = (x-1)(x+1)$ etc, your problem reduces to the form of $(x-a)(x-b)(x-c)(x-d)...(x-z) \ge 0$
Edit: oops I only read the hand-written part! Anyways thanks to the monot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Solving LP from tableau $$\begin{array}{cccccc}
& x1 & x2 & x3 & x4& x5 \\
-4& 2 & 0& -2 & 0& 3\\
3 & 1 & 0 & -1& 1 & 3\\
2 &0& 1 & 0& 0.5 & 2\\
\end{array}$$
When I learned about solving LP represented by the tableau in class, I thought I need to select the column that ha... | $$\begin{array}{cccccc}
& x1 & x2 & x3 & x4& x5 \\
-4& 2 & 0& -2 & 0& 3\\
3 & 1 & 0 & -1& 1 & 3\\
2 &0& 1 & 0& 0.5 & 2\\
\end{array}$$
To begin with a bfs
$$\begin{array}{cccccc}
& x1 & x2 & x3 & x4 & x5 \\
-10& 0 & 0 & 0 & -2 & -3 \\
3 & 1 & 0 & -1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/112816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Finding derivative of $\sqrt[3]{x}$ using only limits I need to finding derivative of $\sqrt[3]{x}$ using only limits
So following tip from yahoo answers: I multiplied top and bottom by conjugate of numerator
$$\lim_{h \to 0} \frac{\sqrt[3]{(x+h)} - \sqrt[3]{x}}{h} \cdot \frac{\sqrt[3]{(x+h)^2} + \sqrt[3]{x^2}}{\sqrt[3... | Here is a hint: Use the identity $(a^3-b^3)=(a-b)\cdot(a^2+ab+b^2)$ with $a$, $b$ being suitable cube roots. Otherwise, the method is similar to the one you tried.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/112865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Show $4 \cos^2{\frac{\pi}{5}} - 2 \cos{\frac{\pi}{5}} -1 = 0$ Show
$$4 \cos^2{\frac{\pi}{5}} - 2 \cos{\frac{\pi}{5}} -1 = 0$$
The hint says "note $\sin{\frac{3\pi}{5}} = \sin{\frac{2\pi}{5}}$" and "use double/triple angle or otherwise"
So I have
$$4 \cos^2{\frac{\pi}{5}} - 2 (2 \cos^2{\frac{\pi}{10}} - 1) - 1$$
$$4 \... | Here is an alternate approach.
Using de Moivre's Formula, we get for $\theta=\frac{\pi}{5}$
$$
0=(\cos(\theta)+i\sin(\theta))^5+1\tag{1}
$$
Looking at the real part of $(1)$ yields
$$
\begin{align}
0
&=\color{red}{\cos^5(\theta)}\color{green}{-10\cos^3(\theta)\sin^2(\theta)}\color{blue}{+5\cos(\theta)\sin^4(\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/113466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
A problem about parametric integral How to solve the following integral.
$I(\theta) = \int_0^{\pi}\ln(1+\theta \cos x)dx$ where $|\theta|<1$
| Note that the integrand always changes sign at $x=\frac\pi2$ for $\theta\ne0$.
In fact, this is an even, nonpositive function,
since $\cos(\pi-x)=-\cos x$ and since, for $r=\theta\cos x$,
$|r|<1$ and $\ln(1+r)+$ $\ln(1-r)=$ $\ln(1-r^2)<0$ $\implies$
$$
\eqalign{
\int_0^\pi~\ln\big(1+\theta\,\cos\,x\big)\;dx &=
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/114401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
How were trigonometrical functions of $\dfrac{2\pi}{17}$ calculated? I know they were calculated by Gauss, but how? Is there a method for calculating them?
| Let $\omega = e^{2 \pi i/17}$. Since $3$ is a primitive root mod $17$, i.e. a generator of the multiplicative group of nonzero integers mod $17$, write $R_j = \omega^{3^j}$ for $j = 0, 1, \ldots, 15$. These and $1$ are the $17$'th roots of unity.
For $2^j \le i < 2^{j+1}$ let $x_i = \sum_{k \equiv i \mod 2^j} R_k$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$. Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.
Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$
I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyo... | $x$ divides $x^2 + y^2 + 1$ implies $y^2 = ax - 1$.
Then $y$ divides $x(x+a)$
Case 1 - $y$ divides $x$, so $x = by$.
$$1/b + b + 1/(by^2) = k$$.
$$b=x=y=1$$
$$k=3$$
Or, $b=x=2$, $y=1$, $k=3$
Case 2 - $y$ divides $x+a$, so $y = -a \text{ mod } x$, $y^2 = 1 \text{ mod } x$.
$$x^2 + y^2 +1 = 2 \text{ mod } x$$
$x$ is $1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/115272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 5,
"answer_id": 4
} |
How do I integrate this? How do I integrate $\displaystyle\int (x^2 + 2)\sqrt{1-x} \; dx$ ?
I have feeling substitution might be used, but I just can't put my finger on it...
Thank you.
| Note that $$x^2+2=(1-x)^2-2(1-x)+3,$$ which implies that
$$(x^2+2)\sqrt{1-x}=(1-x)^\frac{5}{2}-2(1-x)^{\frac{3}{2}}+3(1-x)^{\frac{1}{2}}.$$
Therefore, let $u=1-x$, we have $dx=-du$, which implies that
$$\int (x^2+2)\sqrt{1-x}dx=-\int u^\frac{5}{2}-2u^{\frac{3}{2}}+3u^{\frac{1}{2}}du$$
$$=-\frac{2}{7}u^{\frac{7}{2}}+\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Divisibility of integers Let $n > 1$ be an integer. Then $2^n - 1\nmid 3^n - 1$. I don't know how to prove it. Can anybody help me, please?
In general, for a fixed positive integer $a > 1$, has $a^n - 1|(a +1)^n - 1$ any integer solutions?
| As @AQP said, if $n$ is even then $3\mid 2^n-1$ so $2^n-1\nmid 3^n-1$.
If $n=2k-1$
then
$2^n-1 \equiv 1 \pmod{3}$ so $2^n-1$ is a quadratic residue mod 3.
$3(3^n-1)=3^{2k}-3$ so $2^n-1 \mid 3^n-1$ would require that $3^{2k}\equiv 3 \pmod{2^n-1}
$,
i.e. that 3 is a quadratic residue mod $2^n-1$.
But $2^n-1\equiv 3 \pmo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/116978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
powers of $\frac{1+\sqrt a}2$ For any a which is not a perfect square, let $x=\frac{1+\sqrt a}2$.
$x^n$ can be written uniquely as $b_nx+c_n$, where b and c are rational.
Apart from $a=0, a=1, a= 1 \pm 2^m$ for $m>2$, are there any other values of $a$ for which $b$ or $c$ is an integer for infinitely many $n$? If not... | If $a \equiv 5$ ($\bmod 8$) then this happens infinitely many times. This follows from the relation $x^2 = x + \tfrac{a-1}{4}$ and $\tfrac{a-1}{4}$ is odd. Suppose $x^n = b_nx + c_n$ for integers $b_n,c_n$ then
$$
x^{n+1} = (b_n+c_n)x + b_n\frac{a-1}{4}
$$
So $b_n \equiv 1, 1, 0, 1, 1, 0, \dotsc$ ($\bmod 2$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/119981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How I can find the value of $abc$ using the given equations? If I have been given the value of
$$\begin{align*}
a+b+c&= 1\\
a^2+b^2+c^2&=9\\
a^3+b^3+c^3 &= 1
\end{align*}$$
Using this I can get the value of
$$ab+bc+ca$$
How i can find the value of $abc$ using the given equations?
I just need a hint.
I have tried ... | If $a,b,c$ solve the equation $x^3+mx^2+nx+p=0$ then you know that $S_3+mS_2+nS_1+3p=0$, where $S_i=a^i+b^i+c^i$. From the sum you find who $m$ is. The expression of $n$ is $ab+bc+ca$. You can find $p$ substituting all the values in the equation. Then you can find the product, which is $-p$ and eventually solve the equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/120536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
} |
The solution set of the equation $|2x - 3| = - (2x - 3)$ The solution set of the equation $\left | 2x-3 \right | = -(2x-3)$ is
$A)$ {$0$ , $\frac{3}{2}$}
$B)$ The empty set
$C)$ (-$\infty$ , $\frac{3}{2}$]
$D)$ [$\frac{3}{2}$, $\infty$ )
$E)$ All real numbers
The correct answer is $C$
my solution:
$\ 2x-3 = -(2x-3)$... | $$\left | 2x-3 \right | = -(2x-3)$$
$let$, $t= 2x-3$
$$\left | t \right | = -t$$
$$t=<0$$
$$2x-3=<0$$
$$x \in \left(-\infty, \frac{3}{2}\right]$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/121240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
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.