Questions tagged [binomial-theorem]

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

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1answer
8 views

How does this vandermonde identity proof works?

Recently I have taken a look at this vandermonde identity proof Inductive Proof for Vandermonde's Identity?. \begin{align*} \binom{m + (n+1)}r &= \binom{m+n}r + \binom{m+n}{r-1}\\ &...
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1answer
13 views

binomial coefficients with the given function

A sequence is given as $f(n)=\dfrac{1+dx+ex^2}{1+ax+bx^2+cx^3}$.we have to calculate value of $f(n)$.we know the value of $a,b,c,d,e,n$. and we have to find $f(n)$. I think the answer should be from ...
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0answers
27 views

Is the Binomial Theorem extendable to other fields?

I very well know that the Binomial Theorem can be extended from an integer power, to a rational exponent, any real number and perhaps to the complex field with $z^w=e^{w \cdot ln(z)}$ or $(x+iy)^{u+iv}...
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2answers
60 views

Prove using binomial formula $9 \mid 10^k - 1$ for $k \in \mathbb{N}$

Prove using binomial formula $9 \mid 10^k - 1$ for $k \in \mathbb{N}$ I am aware of similar answer to this question here and here however my query is about the manipulation on the binomial formula in ...
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15 views

Limit of sum involving binomial coefficients

Let $\alpha\in (0,1)$, $\delta \in (0,1)$, $\nu \in \mathbb{Z}^{+}$ with $\varphi = \frac{\delta(1-\alpha)}{1-\delta(1-\alpha)}\frac{1}{\nu}$. To what does the following limit converge? $\lim_{n \...
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2answers
40 views

How do I prove that ${n-1 \choose k} - {n-1 \choose k-1}$ is equal to ${n-2k\over n}{n \choose k}$? [closed]

I really need help as I am really struggling here on what to do. $${n-1\choose k}-{n-1\choose k-1}=\frac{n-2k}n{n\choose k}$$
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2answers
64 views

Why does $(6+i)^3 = 198+107i$?

When I expand it I get $i^3+18i^2+108i+216$. How does one go from that to $198+107i$. I noticed that the 1st term of the 2nd equation is = to the 4th term of the 1st equation - the 2nd term of the 1st ...
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1answer
40 views

How to apply Multinomial Theorem here?

I know that if $I_1, \dots, I_n$ are finite index sets and for every $i \in \bigcup_{j=1}^n I_j$ there exists $v_i \in \mathbb{R}$, then holds $$ \prod_{j=1}^n \sum_{i \in I_j} v_i = \sum_{(i_1, \dots,...
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1answer
60 views

Finding the number of terms in $(1+x)^{100}+(1+x^2)^{100}+ (1+x^3)^{100}$

Find the number of terms in the expansion in $$(1+x)^{100}+(1+x^2)^{100}+ (1+x^3)^{100}$$ Note that $(1+x)^{100}$ gives 101 terms ranging from x^0 to x^100, since all the terms are positive we have ...
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0answers
47 views

Upper bound on the sum with binomial coefficients

Let $0\leq x\leq i\leq y\leq n$, where $n \in N$. Find an upper bound of the sum $$ \sum_{i=x}^y{i+x-1 \choose x-1}{n+y-i \choose y-i} $$
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49 views

integral of a function with exponent

Can anybody help me to solve the following integral? $\int_{0}^{\infty}(1-(ax+1)^{(1-y)} e^{-\frac{x}{b}})^M dx$ It confuses me, because there is this Mth power of overall function. Do I need to use ...
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33 views

What does this binomial coefficient property mean?

The following binomial coefficient property, $\displaystyle \sum_{s=0}^{n-1}\binom{k+s}{k}=\binom{n+k}{k+1}$, is described in the book Mathematical Methods for Physics and Engineering—by K.F.Riley—...
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2answers
45 views

Finding even degree of a polynomial using even function trick

The sum of the coefficents of all even degree terms in x in the expansion of $(x + \sqrt{x^3 - 1})^6 +( x - \sqrt{x^3 -1})^6 $ , ($x>1$) is equal to: ans:24 I tried to do this by the trick to ...
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1answer
45 views

I got a binomial question and when trying to solve for p and q I reached $x^8$. Am I wrong?

$\left(px-\frac qx\right)^8 $ fifth term is $5670 $ $p-q=2 $ $p$ and $q$ are positive using this info i got: $$x^{8}+560x^{7}+1680x^{6}+2240x^{5}+1120x^{4}-5670=0 $$ i am unable to ...
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65 views

Proof that $a^n+b^n+c^n+\dots$ is divisible by $a+b+c+\dots$ for $a, b, c,\dots$ a finite AP and $n$ an odd natural number.

I was thinking of approaching via the identity $a^n+b^n$ being divisible by $(a+b)$ for $n$ odd and then manipulating the differences of the AP terms. However, it didn't seem trivial. I reckon this ...
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1answer
37 views

Binomial theorem question (How did we get the following equality?)

I'm not clear as to how we get the following equality: $$\dfrac{(x+1)^p - 1}{x} = x^{p-1} + {p \choose 1}x^{p-2} + \cdots + {p \choose p-2}x + {p \choose p-1}. $$
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43 views

Misprint(s) In “Combinatorial Identities” by Riordan?

On page 38 in the exercises I was able to confirm that $$a_{kj}(x)=(x+1)\dots(x+k-1)a_{k-1,j}(x)+x^{k-1}a_{k-1,j-1}(x+1).$$ For $k\ge{}j$ the author defines $b_{kj}(x)$ via $$a_{kj}(x)=(x+j+1)^{k-j-1}(...
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0answers
30 views

Trace of $(A+B)^n$ with B an involutory matrix

Consider two matrices $A$ and $B$ that do not commute, so that the binomial theorem does not apply. However, one of them (say $B$) is an involutory matrix, meaning that $B^2 = I$. I am wondering ...
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2answers
58 views

An extended Freshmans dream [closed]

If $p$ is prime, then $(x+y)^p=x^p+y^p$ holds in any field of characteristic $p$. (proved with the help of the Binomial Theorem) But now I need to prove that this implies that this is true for any $(a+...
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1answer
24 views

Infinite sum of Combinations

we can write $\frac{1}{4^n}$ as $\frac{1}{2^{2n}}$ then I used Stirling’s approximation for factorial for Simplification, but calculation became complicated, then I tried to use L' hopital's rule ...
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1answer
80 views

Summing a series with binomial coefficients without calculus.

The following problem is from a high school problem set. The students do not know how to integrate yet although they are comfortable with differentiation. The problem is Show that $\displaystyle S:=\...
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2answers
53 views

If $\left(1+x^{2}+x^{4}\right)^{8}={^{16} C_{0}}+{^{16}C_{1}} x^{2}+{^{16} C_{2}} x^{4}+\cdots +{^{16} C_{16}} x^{32}$, how to prove…

If I am given that $$\left(1+x^{2}+x^{4}\right)^{8}={^{16} C_{0}}+{^{16}C_{1}} x^{2}+{^{16} C_{2}} x^{4}+\cdots +{^{16} C_{16}} x^{32},$$ how can I prove that $${^{16} {C}_{1}}+{^{16} {C}_{4}}+{^{16} {...
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1answer
34 views

Show that for every natural number $n$ there is equality - Stirling numbers of the second kind [closed]

Show that for every natural number $n$ there is equality: $$\left\{ n\atop 3\right\} = \frac{3^{n-1}-2^n+1}{2}$$ To prove this equality use equality $${{n+1}\brace{m+1}} = \sum_{k=0}^{n}{n\choose k}...
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1answer
40 views

Rewrite binomial formel

I have the following equation: $$\left(\frac{\sqrt{\sigma^2_1+\sigma_2^2}\cdot a}{\sigma_1 \cdot \sigma_2} - \frac{\mu_1\sigma^2_2 + \mu_2\sigma^2_1}{\sigma_1\sigma_2\sqrt{\sigma^2_1+\sigma^2_2}}\...
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1answer
350 views

Arc Length Integral of $x^x$ from 0 to 1 in closed form.

I was recently trying to compute the arc length of $x^x$ from $0$ to $1$ as follows: $$L=\int_0^1 \sqrt{1+\left(\frac{\text{d}}{\text{d}x}x^x\right)^2} \text{d}x=$$ $$\int_0^1\sqrt{1+x^{2x}(\ln x+1)^2}...
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3answers
90 views

Combinatorial Summation

I'm trying to solve the following question: If $s_n$ is the sum of first $n$ natural numbers, then prove that $$2(s_1s_{2n}+s_2s_{2n-1}+\dots+s_ns_{n+1})=\frac{(2n+4)!}{5!(2n-1)!}$$ This is where I'...
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5answers
100 views

Coefficient of $x^{12}$ in $(1+x^2+x^4+x^6)^n$

I need to find the coefficient of $x^{12}$ in the polynomial $(1+x^2+x^4+x^6)^n$. I have reduced the polynomial to $\left(\frac{1-x^8}{1-x^2}\right)^ n$ and tried binomial expansion and Taylor series, ...
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1answer
23 views

'Binomial Expansion' - How to find value of term in expression with given coefficients?

I have been given this question: "Find the value of $c$ if, in the expansion of $(cx + 2)^3$, the coefficient of $x$ is $24$" To solve this question, I have tried using the 'General Term In ...
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2answers
72 views

Calculating $ \sum_{k=0}{n\choose 4k+1}$

Calculate $$\sum_{k=0}{n\choose 4k+1}$$ This should be an easy and short result but I'm messing up somewhere. What I've done so far is take $f(x)=(1+x)^n$ and with the binomial theorem expand $f(1), f(...
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2answers
56 views

Looking for formula for variation of binomial theorem

Is there variation of the binomial theorem as follows? $$\sum_{i=0}^n {m \choose i} a^{m-i} b^i $$ I am trying to find a formula for it that is a function of $n$ without summation notation. I think ...
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1answer
65 views

Simplifying binomial series

I am stuck with a series that I want to simplify- $${2n+1\choose 0} + {2n+1\choose 1} + {2n+1\choose 2} + \dots +{2n+1\choose n}$$ I think somehow the result $${n\choose 0} + {n\choose 1} + {n\choose ...
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1answer
71 views

Use binomial series for $\sqrt{1+x}$ to find the binomial series for $\frac{1}{\sqrt{1+x}}$

The question specifically asks me to use the binomial series for $\sqrt{1+x}$ to find the series for $\frac{1}{\sqrt{1+x}}$. First, I need to find the derivative of the binomial series $\sqrt {1+x}$ I ...
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1answer
55 views

How to find the coefficient of $x^2$ in $(3-2x) \left(1 + \frac{x}{2} \right)^n$ [closed]

In the expansion of $(3-2x) \left(1 + \frac{x}{2} \right)^n$, the coefficient of $x$ is $7$. Find the value of the constant $n$ and hence find the coefficient of $x^2$. I have no idea how to begin to ...
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1answer
72 views

Solving $(z+i)^n+(z-i)^n=0$

Given a complex number $z$, how can I solve $$(z+i)^n+(z-i)^n=0$$ for z? My attempt I thought that I couldn't possibly convert this into Euler's format. So, I took to Binomial Theorem. $$(z+i)^n+(z-i)...
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1answer
87 views

Length of this expression can take in positive integers including 0

$f(x,y,z) = 5x+6y+8z$ such that $x,y,z>=0$ (numbers) , and f(x,y,z) max is 160 ,we need to find total no possible values this expression $f(x,y,z)$ takes less than than 161 with all x,y,z greater ...
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2answers
77 views

Upper bound for $\sum_{i = 0}^{k-1} {n \choose i} (1 - \varepsilon)^i\varepsilon^{n-i}$

What is tight upper and lower bound for following expression where $0 < \varepsilon < 1$ and $1\leq k \leq n$. $\sum_{i = 0}^{k-1} {n \choose i} (1 - \varepsilon)^i\varepsilon^{n-i}$
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0answers
18 views

Understanding how the binomial theorem is used in weather prediction

I'm researching the use of the binomial theorem for an intro to maths course at my university and I am having a bit of a hard time understanding how the theorem is used in weather prediction. I'm ...
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2answers
49 views

Does the power rule come from the generalised binomial theorem or the other way around?

Remember how we say $lim \frac{\sin{x}}{x}=1$ doesn't come from the L Hospital's rule, because the differentiation of $sin x$ from first principles uses the fact that $lim \frac{\sin{x}}{x}=1$? A ...
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2answers
74 views

Are binomial coefficients defined for $k \in \mathbb{R}$

So I've been trying to learn some basic precalc and I got stumped on definition of combinations namely the following confuses me . I know that The symbol ${n\choose k}$ is read as "$n$ choose $k$....
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1answer
88 views

Confusion on how extended binomial theorem works

So I just finished learning the standard binomial theorem and I've just come across the extended (newtons binomial theorem). As expected I am completely baffled about how it works I do not understand ...
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1answer
41 views

$\sum_{m = 0}^n\binom{n}{2m}(-1)^m = 2^{\frac{n}{2}}\cos\left(\frac{n\pi}{4}\right)$

How can I prove that $S_1(n)=\sum_{m = 0}^n\binom{n}{2m}(-1)^m$ is equal to $2^{\frac{n}{2}}\cos\left(\frac{n\pi}{4}\right)$ using the binomial expansion $(1+z)^n=\sum_{r=0}^n\binom{n}{r}z^r$?
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2answers
210 views

Prove that $\sum_{m=1}^{n} (-1)^{m+1} {n \choose m} \frac{1}{m+1} = \frac{n}{n+1}$

I'm trying to prove that: $$\sum_{m=1}^{n} (-1)^{m+1} {n \choose m} \frac{1}{m+1} = \frac{n}{n+1}$$ I've tried to prove this by induction and directly, without luck. Any help would be appreciated.
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1answer
31 views

Newton's Binomial Theorem with $n<0$

I was watching this video from Veritasium (https://www.youtube.com/watch?v=gMlf1ELvRzc) where one of the points he brings is applying the Binomial Theorem with $n<0$. To clarify: $$ (1+x)^n = \frac{...
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4answers
54 views

Finding the coefficient of $x^2$ in an expansion

I was given this question as a practise assignment and I am unsure of my answer. The coefficient of $x^2$ in the expansion of $(x+\frac{1}{ax})^8$ is 7. Find the possible value of $a$. I did $(x+\frac{...
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2answers
69 views

The coefficient of $x^2$ in the expansion of $\left(x^3+2x^2+x+4\right)^{15}$

To find the coefficient of $x^2$ in the expansion of $\left(x^3+2x^2+x+4\right)^{15}$. I saw a solution to this problem in which the above expression was differentiated 2 times and they put x=0. I ...
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0answers
48 views

Binomial theorem related prove [duplicate]

Let n be any positive integer. Prove that $\sum\limits_{k = 0}^m {\frac{{\left( {\begin{array}{*{20}{c}} {2n - k}\\ k \end{array}} \right)}}{{\left( {\begin{array}{*{20}{c}} {2n - k}\\ n \end{array}} \...
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0answers
27 views

prove using binomial theorem $\left(1+\frac{1}{n}\right)^n = 1 + \sum^{n}_{k=1}\left[\frac{1}{k!}\prod^{k-1}_{r=0}\left(1-\frac{r}{n}\right)\right]$ [duplicate]

Using the binomial theorem, prove the following equation using induction. Please have a look at my current approach, as I welcome feedback on tackling this equation. For, I'm currently experiencing ...
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2answers
179 views

Binomial theorem from first principles?

I suppose I'll give a little context to this... I was at first really excited by the prospect that both Bernoulli and Taylor's versions of $e^x$ actually ammount to the same thing (where, when you ...
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2answers
60 views

If $x^n=a_0+a_1(1+x)+\ldots+a_n(1+x)^n=b_0 + b_1(1-x)+\ldots+b_n(1-x)^n$, then for $n=101$, find $(a_{50},b_{50})$

My teacher told me add 1 on both sides of the equation, given $$1+x^n=(a_0+1)+a_1(1+x)+\ldots+a_n(1+x)^n$$ But I don’t see how that’s useful in anyway. Am I misunderstanding the hint? What needs to be ...
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1answer
36 views

Explanation for $k{n \choose k - 1} = (k-1){n \choose k - 1} + {n \choose k - 1}$

Does anyone have a good explanation for why this makes sense? I am trying to remember this fact in a proof but simply can't seem to make a good explanation for it. NOTE: I have already shown that $k{...

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