For questions related to the binomial theorem, which describes the algebraic expansion of powers of binomials.

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0
votes
1answer
39 views

Multiplication of 2 sums that equal another multiplication of 2 sums

I have been trying to prove a formula of mine and i come across something very interesting, well to me it is. If the formula is correct, it states that: $$ \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_1 ...
0
votes
1answer
48 views

Sum on integration and binomial theorem.

If $(1+x)^n = \sum_{r=0}^n \binom{n}{r}x^r$ and $$\sum_{r=0}^n \frac{(-1)^r}{(r+1)^2} \binom{n}{r} = k\sum_{r=0}^n \frac{1}{r+1}$$ Then prove that $$k=\frac{1}{n+1}.$$
6
votes
6answers
117 views

What are the last two digits of $77^{17}$?

I'm trying to solve current task referenced the following but I stuck at $\displaystyle77^{17}\equiv x\pmod{100}$. As it is described on above link it uses Binomial Theorem. But I read a lot about the ...
-1
votes
2answers
23 views

Find the remainder of the following using binomial theorem? [closed]

Find the remainder when $$ 5^{5^{5^{5...}}}$$ (24 times 5 ) is divided by 24 using binomial theorem ? Answer to the question is 5 .
2
votes
1answer
82 views

How to find coefficient of $x^{12}$ in the expansion of $(1+x+x^2+x^3+…+x^n)^4$

How to find coefficient of $x^{12}$ in the expansion of $(1+x+x^2+x^3+...+x^n)^4$ I tried this : Since $(1+x+x^2+x^3+...+x^n)$ is in GP its sum will be $(x^{n+1}+1)(x-1)^{-1}$ now ACQ we have to ...
0
votes
0answers
27 views

Show $\sum_{ r = 0}^{n} {(\binom n r)}^{2} = \frac{(2n)!}{(n!)^{2}}$ [duplicate]

How do we show that this identity holds for any n? Any hints or solutions? Show $\sum_{ r = 0}^{n} {(\binom n r)}^{2} = \frac{(2n)!}{(n!)^{2}}$
2
votes
1answer
69 views

Prove limit converges in definition of $e.$

I've looked up several related questions, but they do not answer what I am looking for. Please give link if this is a duplicate. What I eventually want to know is why ...
3
votes
2answers
55 views

$1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}$

The inequality $1+\sqrt[3]{e^{2a}}\sqrt[5]{e^{b}}\sqrt[15]{e^{2c}} \leq \sqrt[3]{(1+e^{a})^2}\sqrt[5]{1+e^{b}}\sqrt[15]{(1+e^{c})^2}$ is true for all $a,b,c\in\mathbb{R}$? I've tried to use the ...
2
votes
1answer
16 views

Binomial expansion in descending power

For example, find, in ascending powers of $x$, the first three terms in the expansion of $(2+5x)^7$. So, $(2+5x)^7=2^7+\binom{7}{1}(2^6)(5x)+\binom{7}{2}(2^5)(5x)^2$. I've no problem to solve this ...
1
vote
1answer
34 views

prove that $(1+pt)^{p^{r-1}} \equiv 1 \pmod {p^r}$

I need to prove that $(1+pt)^{p^{r-1}} \equiv 1 \pmod {p^r}$ the original question is this: Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$ and I'm following the ...
1
vote
1answer
55 views

Problem with this challenging summation

I'm having some trouble finding the summation of this series. I tried all I could, but in the end the denominator is creating problem. $$ \sum_{r=0}^{n} (-1)^r ...
1
vote
0answers
41 views

Finding a closed form expression for $\sum_{k=\frac {n+2} 2} ^n \binom n k$

Find a closed expression for $\displaystyle\sum_{k=\frac {n+2} 2} ^n \binom n k$, $n$ is even. My attempt: $(1+1)^n = \displaystyle\sum_{k=0} ^ n \binom n k= \sum_{k=0} ^{\frac {n-2} 2}\binom n ...
1
vote
2answers
16 views

Binomial product expansion

I have seen the following mathematical identity in a book: $$ \prod_{i=1}^{N}\left( 1 + ax_i \right)^c = \left( 1 + \sum_{i=1}^{N}{ax_i} + \cdots + \prod_{i=1}^{N}{a^Nx_i} \right)^c $$ Is this a ...
-4
votes
0answers
39 views

Find the sum of this binomial series

Find the sum of the binomial series $$\sum_{r=0}^n (-1)^r \,_nC_r \left[\frac 1{2^r}+\frac{3^r}{2^{2r}} + \frac{7^r}{2^{3r}} + \frac{15^r}{2^{4r}} + \cdots \right]$$ My attempt:I tried converting ...
1
vote
1answer
18 views

Number-theory proof involving the Binomial Theorem

I am trying to prove the following: if $a,b \in \mathbb{Z}$ then $(a+b)^p \equiv a^p + b^p$ (mod $p$), where $p$ is prime. I am recommended to use the fact that $p \choose k$ is divisible by p for $1 ...
0
votes
2answers
39 views

How many ways to distribute $n$ objects into $r$ boxes so that each box have at least $1$ (but no more than $k$) objects?

Example: How many ways are there to distribute 15 fruits to 6 people so that each person has at least 1 fruit but no more than 3? I understand how to do it when we need to make sure that at least ...
-1
votes
2answers
43 views

Factoring $a^{k} - b^{k}$ [duplicate]

I am a bit lost how to factor $a^{k} - b^{k}$. I know it links to the binomial theorem but I can't remember how to do it. Could anyone explain?
1
vote
1answer
29 views

The convergence of the binomial expansion

The binomial expansion of $\frac{1}{z-2}$ is $-\frac{1}{2}\sum^\infty_{n=0} (-1)^n \left(\frac{z}{2}\right)$. Does this converge for $\left|\frac{z}{2}\right|<2$? Does $\frac{1}{z} ...
1
vote
1answer
32 views

Simplifying the binomial expansion

I have $$-\frac{1}{2} \left[1-\frac{z}{2}+\left(\frac{z}{2}\right)^2 -\left(\frac{z}{2}\right)^3+...\right]$$ Why does this equal $-\frac{1}{2} \sum^\infty_{n=0} \left(\frac{z}{2}\right)^n$ and not ...
0
votes
0answers
30 views

Convergence of a binomial expansion

I have the binomial expansion $$1+(-1)(-2z)+\frac{(-1)(-2)(-2z)^2}{2!}+\frac{(-1)(-2)(-3)(-2z)^3}{3!}+...$$ Which simplifies down to $$1+2z+(-2z)^2+(-2z)^3$$ How do I find out if this binomial ...
5
votes
3answers
72 views

binomial expression of a powered term [duplicate]

One answer to a previous question of mine asserted that $$k^2=\binom k2+\binom {k+1}2.$$ I checked that the formula is true. However, it intrigued me. Is there a similar expression for $k^3$? How ...
2
votes
1answer
27 views

Why can the function $f(x)=||A\vec{x}-\vec{b}||^2$ be rewritten as $\vec{x}^tA^tA\vec{x}−\vec{x}^tA^t\vec{b}−\vec{b}^tA\vec{x}+||\vec{b}||^2$

Someone answered a question introducing this transformation of the function $f(x)=||A\vec{x}-\vec{b}||^2$ ; but I cannot get the idea why and how. Looks a bit like a binomial expansion, but I can't ...
0
votes
1answer
36 views

Solving equation with many exponents for same variable

$1.32=5(r + r^4) + 10 (r^2 + r^3) + r^5 + 1$ How would one solve an equation like this? Any help would be appreciated thanks.
4
votes
0answers
38 views

Binomial expansion with negative power?

I tried searching everything but couldn't understand what formula or technique are they using to expand this term. I have attached image of my textbook for this problem. It's written there that we ...
1
vote
1answer
26 views

Falling power of a sum in terms of falling powers of the terms

I am trying to come up with an expression for $(x+y)^{\underline{n}}$ in terms of $x^{\underline{r}}$ and $y^{\underline{r}}$. I tried for $n=2$ and $n=3$ and it looks like binomial expansion holds, ...
0
votes
0answers
42 views

Show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$

To show that the expansion of $(1+x)^n$ by Binomial Theorem is convergent when $x<1$ Let $u_r, u_{r+1}$ represent the $r^{th}$ and $(r + 1)^{th}$ terms of the expansion; then ...
0
votes
0answers
28 views

Does this binomial-like sum have a nicer/simpler/shorter representation?

Using the binomial theorem (or a generalization of the binomial theorem) it is not hard to see that (I found this the book "Statistical Multisource-Multitarget Information Fusion" by Ronald Mahler): ...
1
vote
2answers
35 views

Odd Power terms of binomial theorem proof

I want to acquire all the terms of $(p+q)^n$ where the power of p is odd. Note that $p=1-q$ ($p$,$q$ probabilities) Ex. For $(p+q)^2=p^2+q^2+2pq$ I want to acquire only $2pq$(only term with odd ...
0
votes
4answers
62 views

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$

Find the remainder when $(x+1)^n$ is divided by $(x-1)^3$ I know that \begin{equation*} (1 + x)^n = 1 + nx +\frac {n(n-1)}2!\cdot x^2 +\frac {n(n-1)(n-2)}3! \cdot x^3 +... \end{equation*} ...
2
votes
1answer
48 views

Show that the binomial series satisfies: $(1+x)f'(x)=\alpha f(x)$

If $f(x) = \sum_{n=0}^{\infty}{\alpha\choose n}x^n$, show (without assuming the results of the binomial theorem) that $$(1+x)f'(x)=\alpha f(x)$$ for $|x|<1$ I've already shown that the sum ...
1
vote
1answer
23 views

Show that the following expansion is valid for first $n$ terms

The question is as follows, I've gone ahead and attempted as much as I could of it twice, however the final answer that I'm receiving does not match. I'm omitting the partial fractions steps; if ...
2
votes
2answers
53 views

Prove that $a^{p^2}\equiv a^p\pmod{p^2}$ for every $a\in\mathbb{Z}$

Question: Prove that $a^{p^2}\equiv a^p\pmod{p^2}$ for every $a\in\mathbb{Z}$ First note that when $p$ is prime and $1\leq r\leq p-1$ the binomial coefficient $$\binom{p}{r}=\frac{p!}{r!(p-r)!}$$ is ...
1
vote
0answers
19 views

Lower bounding the binomial summation

For $n>1$ and $~0<p < 1$, can we lower bound the following binomial series in terms of $n$ and $p$ $$\sum_{i=\lceil p n \rceil}^n {n \choose i} (p )^i(1-p)^{(n-i)}.$$ I have posted the ...
3
votes
2answers
100 views

Formal power series coefficient problem

Find the coefficient of: $[x^{33}](x+x^3)(1+5x^6)^{-13}(1-8x^9)^{-37}$ I have figured out that I need to use this identity: $(1-x)^{-k} = \sum\limits_{i>=0} \binom {n+k-1} {k-1} x^n $ But I ...
1
vote
1answer
14 views

Solving product of binomial powers in integers

I have to find $a,b \in \mathbb{Z}^*$ such that $$(1 + x)^5(1+ax)^6 = 1 + bx + 10x^2 + \ldots + a^6x^{11}$$ Now, as far as I can tell for all $a$ there is a solution when $b = 5$, so a solution is ...
0
votes
1answer
28 views

Binomial Expansion

Let $f, g:[0,1]\rightarrow R$ be functions such that $$\left\{\begin{array} & f(x)=g(x)=0 & \Leftrightarrow x=1 \\ f(x), g(x) \neq 0 & \Leftrightarrow x \neq 1\end{array}\right.$$ Now ...
3
votes
4answers
110 views

$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $

$$ \lim_{x\to o} \frac{(1+x)^{\frac1x}-e+\frac{ex}{2}}{ex^2} $$ (can this be duplicate? I think not) I tried it using many methods $1.$ Solve this conventionally taking $1^\infty$ form in no ...
1
vote
3answers
46 views

Summation Of Binomial & Factorial Series

Looking for an explicit formula for the following: $$ S = \sum _{i=j}^n \frac{\binom{i}{j}}{(i+1)!} $$ any Ideas?
0
votes
2answers
35 views

Determine a, b and n in the following expression (binomial theorem backwards)

You are given the following expression: $(ax + by)^{n} = -15120x^{4}y^{3}$ Determine the constants $a$, $b$ and $n$. My attempt to solve this problem is by trying to use the binomial theorem ...
1
vote
1answer
46 views

Binomial Theorem…extra indexed term.

I have the following expression: $$\sum_{i=0}^{n}\binom{n}{i}(2x+1)^{n-i}(-1)^ii!$$ Without the $i!$, the above expression would simply reduce to $(2x)^n$, but is there a way, or method for ...
1
vote
0answers
61 views

Multinomial Theorem for Negative Exponents

Using an analog to Newton's binomial theorem with negative exponents, is it true that $$ \begin{align} \left(\sum_{k=0}^mx^k\right)^{-n} & = \sum_{0\le ...
2
votes
2answers
99 views

The digit at the hundred's place of $33^{33}$

I would want to know how to start with the question. And if you get hung up somewhere there's the answer it's $5$. Any help is appreciated thanks, My approach was to look at the factors to somehow ...
0
votes
1answer
15 views

Suppose that number of mistakes on a page is a Poisson RV and independent. From $n$ pages, find the expected number with no mistakes?

A textbook has $n$ pages. The number of mistakes on each page is a Poisson RV with parameter $\lambda$ and is independent of the number of mistakes on all other pages. What is the expected number of ...
2
votes
2answers
32 views

Deriving the variance of a binomial distribution

I know that the variance of a binomial distribution is the number of trials multiplied by the variance of each trial, but I'm not seeing the derivation of this. Here's my logic so far: For each trial ...
0
votes
1answer
38 views

Binomial Distribution with probability $P$ such that $P$ is Uniformly distributed

A number $P$ is random chosen from the uniform distribution from [0,1]. Then a coin with probability $P$ of getting a head is flipped $n$ times. Let $X$ be the number of heads showing and compute ...
0
votes
1answer
26 views

Finding the $N^{th}$ term In binomial expansion for arbitrary power

How do I find the $n^{th}$ term in the binomial expansion of any index. Ex: Find the $4^{th}$ term of $(1+2x)^{-1/2}$.
1
vote
1answer
59 views

Find the number of irrational terms in the binomial expansion of $(3^{1/5}+7^{1/3})^{100}$

After expanding the above term binomially, I can well guess that the majority terms are irrational, but i'm unable to find any proper method of solving this sum, after repeated trials. Please help ...
1
vote
2answers
53 views

Combinatorial Proof of falling factorial and binomial theorem

For $n,m,k \in \mathbb{N}$ is true: $$(n+m)^{\underline{k}}=\sum^{k}_{i=0}\binom ki \cdot n^{\underline{k-i}} \cdot m^{\underline{i}}$$ I can prove the binomial theorem for itself combinatorically ...
2
votes
1answer
23 views

Division of the Binomial Coefficient

Prove that when p is prime, the binomial coefficient p!/(r!)((p-r)!) is divisible by p with r being greater than or equal to 1 and less than or equal to p-1 . Clearly p! is divisible by p so I cant ...
0
votes
1answer
25 views

Solving the nth number of this recurrence and cleaning it up using the binomial theorem.

Given this recurrence: an = an-1 – an-2 I was told to create a function that would solve for an. I thus came up with $a_n=\frac{\alpha^{n}-\beta^{n}}{i\sqrt{3}}$ Where ...