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

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2
votes
2answers
30 views

Compute Using Binomial Theorem [duplicate]

$$\sum_{k=1}^{10} \binom{10}{k} $$ I know the answer is $2^{10} - 1$ but I don't know how to get to the answer.
6
votes
2answers
135 views

What is the ten's digit of $7^{7^{7^{7^7}}}$

What is the ten's digit of $\zeta=7^{7^{7^{7^7}}}$. I got this question while doing binomial theorem. I think that $7^4=2401$ and we only need $\zeta\pmod{100}$. All I could think of is already ...
0
votes
1answer
67 views

Find the last digit of $(\sqrt{71}+1)^{71}+(\sqrt{71}-1)^{71}$

While teaching binomial expansion, one of my high school students asked me the following question: What is the last digit of $(1+\sqrt{71})^{71}+(1-\sqrt{71})^{71}$? I have absolutely no context on ...
3
votes
7answers
376 views

Expansion of complex equation.

Find the value of $$\left(\frac{-1+\sqrt 3i}{2}\right)^{15} + \left(\frac{-1-\sqrt 3i}{2}\right)^{15}.$$ In general, how do we find the value of expansion of equation of high orders other than ...
4
votes
2answers
56 views

Binomial Theorem of Differentiation? [duplicate]

I noticed that $$\frac{d^{n}}{dx^{n}} f(x)g(x)=\sum_{i=0}^n {n \choose i} f^{(i)}(x)g^{(i)}(x)$$ and it's had me scratching for a little bit. It's easy to see how the cross terms add up but can anyone ...
0
votes
1answer
32 views

Binomial theorem, verify ${\frac12 \choose n+1} = 1/(n+1){n-\frac12 \choose n}(-1)^n(1/2)$ [closed]

I need to verify ${\frac12 \choose n+1} = 1/(n+1){n-1/2 \choose n}(-1)^n(1/2)$ and then verify $${2n \choose n}(1/(2^{2n}))={n-1/2 \choose n}$$ Please help!
0
votes
1answer
29 views

binomial theorem expansion quick question

I'm looking at a proof online to a theorem and some of the steps to the theorem are as follows: let $ x = y+1$ then $$x^p-1 = (y+1)^p -1 = y^p + \sum_{k=1}^{p-1} {p \choose k} y^k$$ $$\iff ...
1
vote
0answers
29 views

coefficients of polynomial and binomial expressions

Let us say we are given a polynomial p(x)=$\sum_k a_k x^k$. In order to find $\sum_k a_k$ we simply need to evaluate p(1), and similarly there are many other tricks. Is there any trick to evaluate ...
2
votes
2answers
60 views

Find the coefficient of $x^4$ in the expansion of $(1 + 3x + 2x^3)^{12}$?

I have not learnt the multinomial theorem yet, and was trying to approach this using the binomial theorem. I divided the terms as $a$ being $(1+3x)$ and $b$ being $2x^3$. I then used $${12\choose ...
1
vote
0answers
24 views

Simplification of an expression sums and products

I am trying to simply the following expression: $ \sum_{\substack{\overline{t}_i , \overline{t}_j \in \{H,L\} \\ \forall j \in N_i}}^{} \sum_{j \in N_i}^{} \alpha_{\overline{t}_i , \overline{t}_{j}} ...
0
votes
2answers
70 views

Determine $(3.12)^5$ to one decimal place.

How many terms in the binomial expansion would be needed to determine $(3.12)^5$ to one decimal place? No calculators allowed by the way. Not really sure how to go about doing it, I could do all the ...
0
votes
1answer
71 views

Proving ${n \choose k}^{-1} = (n+1)\int_0^1 x^k(1-x)^{n-k}\mathsf dx$

Title says it all, I've tried to find the indefinite integral of the right side, got some sort of weird series and got stuck: $$\sum_{i=0}^n-k {n-k \choose i}\cdot{(-1)i\over k+1+i}$$
1
vote
2answers
25 views

Using the binomial theorem to generate a geometric proof of the derivative.

According to wikipedia, if we wanted to prove $$(x^n)'=nx^{n-1}$$ geometrically by creating an $n$-dimensional hypercube $$(x+\Delta x)^n$$ and setting $a=x$ and $b=\Delta x$, we could expand using ...
0
votes
0answers
20 views

How to evaluate the sum $\sum_{k=0}^N {{N}\choose {k}} \exp{(\alpha k^2)}$

If $k^2$ in the exponential were replaced with $k$, it is simple to evaluate by the binomial theorem but is it possible to evaluate this sum in closed form with $k^2$?
1
vote
1answer
66 views

Binomial Series. Product series of coefficients

How to solve this question? Please provide hints only.
0
votes
1answer
17 views

proving $(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ using the binomial theorem

$(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n \{\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn)\}$ this exercise is taken from Apostol's Calculus I (page 45) and it's supposed to be proved by using the binomial ...
0
votes
0answers
31 views

Binomial function [duplicate]

$$S=\sum _{k=1}^n \binom{m+k}{k}$$ As value of n and m can be very big like $10^6 $ or $10^7$, I need S%P where P is a prime. Can anybody help me to derive a closed formula to this sum and also how ...
1
vote
1answer
25 views

Exponents with base close to $1$.

I was just fiddling around with a calculator and calculating powers of numbers really close to $1$ like $1.01,1.001\dots$ trying to find at what value they exceed $2$. This got me thinking if I could ...
0
votes
0answers
15 views

Binomial Expansion without infinity series [Duplicated]

(This is duplicated due do the lack of answers in the previous asking session,because I edited the unreadable math too long that this got lost in "recent questions".I was unable to delete the previous ...
2
votes
1answer
35 views

Binomial Expansion without inifinty series

Variable here is "a" and "b".The question is to simplify the $\sqrt [3] { (\frac{a}{\sqrt{b^3} })\times {\frac {\sqrt{a^6b^2}}{a} }+ \frac{a}{b^2}}$ So these are my steps =$\left(\frac {a^3b} ...
0
votes
2answers
28 views

By the binomial theorem, use this result to show with explanation that the number of subsets of a set $S$ is $2^{|S|}$

Given that $(1+1)^n = 2^n = \sum^n_{k=0} \binom{n}{k}$ by the binomial theorem use this result to show with explanation that the number of subsets of a set $S$ is $2^{|S|}$ I'm really confused. So ...
0
votes
1answer
35 views

Why is my Binomial Theorem not working?

Ok this might seem like a dumb question which I think it is but it is really bugging me. Lets say you have a 1 in 5 chance at something ( doesn't matter what) and you get 2 attempts at it. By basic ...
8
votes
4answers
295 views

Orthogonality for Binomial Coefficients

Could somebody explain to me where these two formulas come from as applications of the binomial theorem? $$\sum_{k=0}^n {n \choose k}(-1)^kk^r=0$$ for non-negative integers $r\lt n$. And ...
0
votes
0answers
24 views

Binominial Theorem proving

As I was trying to understand the proof of Binomial Theorem by induction, I got stuck at this line. What formulas should be used to get from left to right part? Any explanations and answers ...
3
votes
1answer
44 views

Proving combinatorially $\sum_{k=0}^n k \binom n k ^2=n\binom{2n-1}{n-1}$

Prove with a combinatorial proof (story): $\displaystyle\sum_{k=0}^n k \binom n k ^2=n\binom{2n-1}{n-1}$ My attempt: Let's make it easier to work with (it's very easy to show that identity): ...
0
votes
2answers
25 views

Is there limitation when applying binomial theorem?

Problem as title showed. $(a+b)^{-n}$. If $n$ is a positive integer. Can $a$ or $b$ be a complex number? Many thanks in advance.
6
votes
2answers
66 views

Binomal theorem show that $\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\dots=\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+\dots=2^{n-1}$

I'm having some trouble with this question Show that $$\binom{n}{0}+\binom{n}{2}+\binom{n}{4}+\dots=\binom{n}{1}+\binom{n}{3}+\binom{n}{5}+\dots=2^{n-1}$$ Attempt: Expanding $(1+1)^n=2^n$ ...
4
votes
4answers
118 views

What is the story behind ${n+1 \choose k} = {n \choose k} + {n \choose k-1}$? [duplicate]

By exploring the inductive proof from this question I came to the point where I did not understand this step: $${n+1 \choose k} = {n \choose k} + {n \choose k-1}$$ There is a wikipedia article but ...
1
vote
1answer
47 views

Binomial Coefficients

I want to get ahead in my classes and learn Binomial Theorem ahead of time. What I know so far is that the formula below is the Binomial Coefficient: $\binom n k = \frac {n!} {(n-k)!k!}$ and ...
2
votes
2answers
40 views

Power series, part of proof explanation.

I am struggling to understand the red inequality $\color{red}{\leq}$. how one could just change $k$ to $2$, and then change the sum previously going from $2$ to n, to a sum going from $0$ to $n$. ...
2
votes
1answer
26 views

Need an explanation for binomial theorem question

I understand the basic concept of binomial theorem, however once I begin looking at this question I can't seem to wrap my head around why x^0 is equivalent x^16 once you change the form of the ...
0
votes
1answer
44 views

An alternating sum with binomial coefficients is not a prime number

Suppose $n\geq 5$ is an odd positive integer. Prove that $${n \choose 1}-5{n \choose 2}+5^2{n \choose 3}-...+5^{n-1}{n\choose n}$$ is not a prime number. I tried expanding each to see if anything ...
0
votes
1answer
77 views

Find the sum of coefficient of all the integral power of $x$ in the expansion of $\big(1 + 2\sqrt x\big)^{40}$? [closed]

While going through certain question online. This question took a lot of my time. Can anyone please help me with this question!!
0
votes
1answer
21 views

Binomial theorem for vectors

Let $u, v$ be vectors in n-dimensional Euclidean space. Then does the following hold? : $|u+v|^p=|u|^p+p|u|^{p-2}u \cdot v +\dfrac {p(p-1)}2|u|^{p-3}|v|u \cdot v+\dots+|v|^p$
0
votes
1answer
34 views

Simplifying $\binom{n}{k}$ / $\binom{n}{k-1}$

So the question is as follows: Simplify $$ \frac{\binom{n}{k}}{\binom{n}{k-1}}. $$ And this is what I got: $$\begin{align*} \frac{\binom{n}{k}}{\binom{n}{k-1}} &= ...
1
vote
4answers
56 views

Binomial Theorem and Summation [duplicate]

All positive integers $m_1, m_2, n$ satisfy: $$ \sum_{k=0}^n {m_1 \choose k}{m_2 \choose n-k} = {m_1 + m_2 \choose n} $$ Prove using the binomial theorem and the fact that $(1+x)^{m_1}(1+x)^{m_2} = ...
-1
votes
1answer
48 views

How to Taylor expand $(a+b)^n$.

I don't know how to taylor expand $(a + b)^n$ ,can someone send me the proof?
2
votes
1answer
42 views

Multinomial theorem: Number of elements where all coefficients have even powers..

Consider $(a_1+a_2+ \cdots + a_n)^r$. We know that it has $n^r$ elements. I want to calculate the number of elements, where all $a_i$ coefficiants have even powers, i.e. $(a_1+a_2+ \cdots + a_n)^r = ...
0
votes
0answers
17 views

binomial expansion

I have found a paper that approximates the $n$th moment of random variable by $$\mathbb{E}[R^n]=\sum_{n_1=0}^n\sum_{n_2=0}^{n_1}\cdots \sum_{n_{M-1}}^{n_{M-2}} {n\choose n_1}{n_1\choose n_2}\cdots ...
3
votes
1answer
46 views

Using Binomial Theorem for proofs. [duplicate]

I need to prove that: $$0 = \sum_{k=0}^n\binom{n}{k}(−1)^k$$ Any ideas, please?
2
votes
9answers
137 views

If $n\ge2$, Prove $\binom{2n}{3}$ is even.

Any help would be appreciated. I can see it's true from pascal's triangle, and I've tried messing around with pascal's identity and the binomial theorem to prove it, but I'm just not making any ...
8
votes
6answers
354 views

How to find closed form of a binomial series.

When working on a problem, I needed to find the closed form of the infinite sequence: $$1 - 2x + 3x^2 - 4x^3 + \cdots$$ I struggled with this for a while and eventually found, through the Internet, ...
0
votes
1answer
22 views

Help with Binomial Theorem: Greatest coefficient

my textbook said to determine the greatest coefficient in a binomial expansion $ (a+b)^n $ we can use the inequality: \begin{align} \frac{n-k+1}{k} \cdot \frac{b}{a} \geq 1 \end{align} Then solve ...
1
vote
2answers
35 views

Negative binomial theorem

I have been supplied with a combinatorical proof based on the n'th power, however I am trying to prove this by induction. I have no problem with the base case, or assuming that n=N. However, for ...
-1
votes
2answers
49 views

What is the proof that the two below are equal? [duplicate]

Can you help me to prove the following equality ?
0
votes
1answer
43 views

How to compute $\displaystyle\sum_{k\equiv 1\!\!\pmod{\!4}}\!\!\binom{2014}{k}$?

I have to compute: $S=\binom{2014}{1}+\binom{2014}{5}+\binom{2014}{9}+...+\binom{2014}{2009}+\binom{2014}{2013}$ Could someone help me ?
0
votes
0answers
23 views

How to compute $\int_{-\infty}^{\infty} [(\frac{v}{i+1}+1)^2+\frac{1}{2}]^{-n-2} e^{-a v^2 } \mathrm{d}v$

How to compute the following integral? $$\int_{-\infty}^{\infty} [(\frac{v}{i+1}+1)^2+\frac{1}{2}]^{-n-2} e^{-a v^2 } \mathrm{d}v$$ in which, $a$ is a positive real number, $n$ is positive integer and ...
1
vote
4answers
109 views

How to expand $(x+y)^{-n}$?

How to expand $(x+y)^{-n}$ by binomial theorem, where $n$ is a positive integer? Is there any limitation for $x$ and $y$? If it can be expanded, how to compute the coefficients? Many thanks in ...
0
votes
1answer
32 views

Finding the coefficient of $x^{r}$

Find the coefficient of $x^{r}$ in the expression - $$(x+3)^{n-1} +(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^{2}+....+(x+2)^{n-1}$$ Attempt- I tried to write it as a binomial expansion but the binomial ...
0
votes
1answer
54 views

Use of Multinomial theorem.

I have the next identity which I want to prove. $$(\sum_{j}k_j^2)^{s} = \sum_{b_1+\ldots+ b_n =s} \prod_j k_j^{2b_j}$$ Obviously I need to use the Multinomial theorem, but how to procceed from ...