5
votes
3answers
654 views

Preventing “proof by homework”?

I am doing problem 3d in the Prologue of Spivak: Prove $(a+b)^n = a^n + {n\choose1}a^{n-1}b + {n\choose2}a^{n-2}b^2 + ... + {n\choose n-1}ab^{n-1} + b^n$ I feel like my proof could pass off as ...
2
votes
5answers
59 views

How to find the coefficient of $x^8$ in $\prod\limits_{i=1}^{10}{\left(x-i\right)}$?

How to find the coefficient of $x^8$ in $(x-1) (x-2) . . .(x-10)$. Is there any general formula to solve this kind of problems?
3
votes
2answers
67 views

Simplify a triple sum

I need to find a closed form for this summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m}\choose{k}}}{j{m\choose j}}r^{k-j+i}$$ I posted this a long time ago, but today I found out ...
5
votes
2answers
106 views

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$ My work: $\sqrt8=\bigg(1-\dfrac12\bigg)^{-\frac32}$ Now, I suppose there is some "binomial ...
1
vote
1answer
40 views

Problem with raising parentheses to powers

Simply math question, lets say I have $(2x^2)^3$.Is this equal to $8x^6 , 2x^5, 2x^6$, or $8x^5$ ? It is a simple problem but what confuses me is do if I multiply the coefficient separately from the ...
5
votes
7answers
293 views

If $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$.

If $x>0$ and $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$. Is there some other way to do find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
0
votes
0answers
29 views

Can this binomial coefficient term be simplified?

Can this be simplified? $$\binom{n}{k}\binom{k}{j}2^{-k}$$ assuming $k \le n$ and $j \le k$? I've tried expanding it in to factorials, but other than a $k!$ term, nothing seems obvious. ...
0
votes
2answers
154 views

Binomial Theorem Practical Problem

I have been studying the 'Binomial Series', Chapter 16, Pg.125 within the Engineering Mathematics Book by John Bird. After completing this section I have attempted to complete the exercises for ...
3
votes
2answers
71 views

$2^n=C_0+C_1+\dots+C_n$

Could anyone give me hints for this one? $2^n=C_0+C_1+\dots+C_n$ Thats all I can say and I know tricks like integrating or differentiating both side and then put $x=1$ or what ever we need. but ...
2
votes
2answers
147 views

Use induction and Newton's binomial formula to show that $\binom{n}{0}+\binom{n}{1}+\cdot+\binom{n}{n}=2^n, \forall n\in \mathbb N$ [duplicate]

Use induction and Newton's binomial formula to show that: $ i)$ $ \binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n}=2^n, \forall n\in \mathbb N$ $ ii)$ ...
2
votes
0answers
173 views

How to calculate this triple summation?

I need to calculate the following summation: $$\sum_{j=1}^m\sum_{i=j}^m\sum_{k=j}^m\frac{{m\choose i}{{m-j}\choose{k-j}}}{k\choose j}r^{k-j+i}$$ I do not know if it is a well-known summation or not. ...
2
votes
1answer
44 views

Proving Bernoulli's Inequality for $h<0$

I'm answering question 19 of chapter two of Spivak's Calculus and I can't seem to think of a way of doing it. I don't want to look up the answer so I thought I'd ask for a hint as to the general ...
0
votes
2answers
87 views

How can I compute this sum of binomial

Is there any way to compute the following sum: $\displaystyle{ \sum_{\ell = {n + 1 \over{\vphantom{\LARGE A}2}}}^{n}{n \choose \ell}5^{n - \ell}}$ where $n$ is odd. Thank you.
0
votes
0answers
54 views

Close formula for triple sum binomial coefficient

I need to compute the following sum or to find a lower and upper bound that limit the sum: $\sum_{l=\frac{n+1}{2}}^n \binom{n}{l} \sum_{t=0}^{n-l} \binom{l}{t} 2^{l-t} \sum_{m=t}^{n-l} \binom{n-l}{m} ...
2
votes
1answer
70 views

Sum of series ${n\choose 2a}{a\choose 0}+ {n\choose {2a+2}}{{a+1}\choose 1} + {n\choose {2a+4}}{{a+2}\choose 2} + \ldots$

I wanted to check the rationality of the cosine function for some rational multiples of $\pi$. And I found out that, $\cos(n \cdot\arccos x)$ generates a polynomial in $x$ whose co-efficients have the ...
2
votes
1answer
42 views

Alternating binomial sum with intervals of two

Fix integer $n\geq 1$. Consider the number $$1-\binom{n}{2}+\binom{n}{4}-\binom{n}{6}+\cdots$$where the sum continues as long as the lower number in the binomial is $\leq n$. Is there a way to ...
1
vote
1answer
53 views

Simplification of Binomial Expansion.

How $$(x+h)^n-x^n=nhx^{(n-1)}\text{ ?}$$ My attempt : $$ \begin{align} (x+h)^n-x^n & =nhx^{(n-1)} \\[8pt] & =\left[\sum_{k=0}^{n}\binom{n}{k}x^{(n-k)}h^k\right]-x^n \\[8pt] & = ...
1
vote
1answer
50 views

Simplify Mathematical Expression

Can someone help me to simplify the following expression? I can assume b is small and $0<b<1$. $(C^{N}_{i})$ is the binomial coefficient. $$A=[(1-b)^{N} + \sum^{N}_{i=1} (C^{N}_{i} - 2 ...
-2
votes
1answer
98 views

Is this just a version of the binomial theorem?

I asked a question related to it and found something interesting (at least that is what I think)... Here is the link to the original question: What is the pattern of this sequence? I went through a ...
3
votes
3answers
93 views

Find the coefficient using binomial theorem.

What is the coefficient of $x^{20}$ in the expression: $$(x+1)^{10}.(x^2 -1)^8$$
5
votes
3answers
233 views

Dividing factorials is always integer

Is there a simple way to show that $$n!\over r!(n-r)!$$ is always an integer?
4
votes
1answer
77 views

Binomial theorem for prime exponent

Could you explain to me why for prime $p$ we have the following? $$(x+y)^p - (x^p + y^p)= x^p + \binom{p}{1}x^{p-1}y + \binom{p}{2}x^{p-2}y^2 + \binom{p}{p-1}xy^{p-1} + y^p.$$ I found it here: ...
-2
votes
2answers
108 views

Calculation of binomial sum $\displaystyle \sum_{r=1}^{n}r.\binom{n}{r}x^r.(1-x)^{n-r} = \;\;?$ [closed]

How can I calculate $$\displaystyle \sum_{r=1}^{n} r \binom{n}{r}x^r (1-x)^{n-r} =\;\; ?$$
5
votes
4answers
193 views

Proving $\binom{2n}{n}\le 4^n$ for all $n$ by smallest counterexample

Prove $$\binom{2n}{n}\le 4^n$$ for all natural numbers $n$ by smallest (minimal) counterexample. My attempt: First, $$\binom{2n}n = \frac{(2n)!}{(n!)^2} \le 4^n\;.$$ We know that $x\ne 0$ because ...
6
votes
3answers
234 views

How many solutions are possible to this equation?

Given $$A+2B+3C=N $$ where $N$ is a given positive integer. $A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$. How many solutions will be there to this equation?
7
votes
1answer
167 views

Closed-form expression for $\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}$?

Wolframalpha tells me that $$\sum_{n=1}^{k} (-1)^{n+1}n^2(n^2-1)\binom{2k}{k-n}=0$$ for $k>2$ However I have not been able to come up with a proof and I simply don't see how to do it. Does anyone ...
2
votes
3answers
186 views

Can $n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$ be derived from the binomial theorem?

Can this identity be derived from the binomial theorem? $$n(n+1)2^{n-2} = \sum_{i=1}^{n} i^2 \binom{n}{i}$$ I tried starting from $2^n = \displaystyle\sum_{i=0}^{n} \binom{n}{i}$ and dividing it ...
0
votes
1answer
69 views

How did my textbook conclude this proof?

http://i.imgur.com/goUlA.png At the very last step, highlighted in red, it states that if this, then that sort of, and I'm not sure how those comparisons explain the issue of proving Pascal's ...
0
votes
1answer
96 views

Could anyone explain how my textbook did this simplification?

The book is talking about proving Pascal's triangle increases until the middle, until which point it decreases. Theorem 4.2 refers to the Multiplicative formula I just don't understand how that ...
0
votes
2answers
839 views

Calculate reciprocal square root via binomial expansion

I have to use a binomial expansion to evaluate $1/\sqrt{4.2}$ to $5$ decimal places. The answer from a calculator is $0.48795$ but I get $0.48202$, so I'm doing something wrong. I've also checked my ...
2
votes
2answers
163 views

Simplify $\sum_{i=0}^{n-1} { {2n}\choose{i}}\cdot x^i$

I am trying to simplify an expression involving summation as follows: $$\sum_{i=0}^{n-1} { {2n}\choose{i}}\cdot x^i$$ where $n$ is an integer, and $x$ is a positive real number. At a first ...
2
votes
2answers
2k views

Find the coefficient of $x^3y^2z^3$ in the expansion $(2x+3y-4z+w)^9$

The exercise says: In the expansion $(2x+3y-4z+w)^9$, find the coefficient of $x^3y^2z^3$. The formula to find the coefficient of $x_1^{r_1}x_2^{r^2}\dots x_k^{r_k}$ in $(x_1+x_2+\dots+x_k)^n$ ...
2
votes
2answers
303 views

Does this qualify as a proof? (Spivak's 'Calculus')

I'm working through Spivak's 'Calculus' at the moment, and a question about series confused me a bit. I think I have the solution, but I'm not sure if my "proof" holds. The question is: Prove ...
4
votes
0answers
126 views

How to transform series of series into series

I need to prove this equation. $$ \sum_{k=0}^{i-2} \left( e \space α(k+1)\space\frac{(-1)^{i+k+2}}{(i-k-2)!} \right) = \sum_{k=0}^{i-2} \frac{(i-k)^k}{k!} \space e^{i-k} (-1)^k\space where,\space α(i) ...
2
votes
2answers
185 views

simplify $(a_1 + a_2 +a_3+… +a_n)^m$

How to simplify this best $(a_1 + a_2 +a_3+... +a_n)^m$ for $m=n, m<n, m>n$ I could only get $\sum_{i=0}^{m}\binom{m}{i}a_i^i\sum_{j=0}^{m-i}\binom{m-i}{j}a_j ... $
3
votes
2answers
128 views

Why is the binomial coefficient related to the binomial theorem?

The binomial coefficient basically provides the number of ways to choose a set of $k$ from $n$ sets. To me, it can be considered the number of unique ways to pick $k$ amount of "cards" from a deck of ...
7
votes
4answers
320 views

Showing that $\lceil (\sqrt{3} + 1)^{2n} \rceil$ is divisible by $2^{n+1}$.

I have a question which has fluxommed me and my pals for the past few days. Any help or solution is welcome Show using Binomial theorem that the integer just after $(3^{1/2} + 1)^{2n}$ is divisble ...
4
votes
5answers
214 views

The sum of the coefficients of $x^3$ in $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$

I know how to solve such questions when it's like $(x+y)^n$ but I'm not sure about this one: In $(1-\frac{x}{2}+\frac{1}{\sqrt x})^8$, What's the sum of the coefficients of $x^3$?
7
votes
3answers
426 views

Hard elementary combinatorics problem

How does one compute (without brute force) the smallest integer $n$ such that $\binom{2n}{1}(-3)^0 + \binom{2n}{3}(-3)^1 + \binom{2n}{5}(-3)^2 + \cdots + \binom{2n}{2n-1}(-3)^{(n-1)} = 0$?
3
votes
5answers
448 views

Binomial theorem application

I have a question about the bonomial theorem, and in specifically, a question that I want help on. I have worked out the answer, but by manually expanding each and every alternative. However, I ...
1
vote
3answers
1k views

Calculate the expansion of $(x+y+z)^n$

The question that I have to solve is an answer on the question "How many terms are in the expansion?". Depending on how you define "term" you can become two different formulas to calculate the terms ...
2
votes
3answers
414 views

Water and wine mixing problem

This is a well-known problem involving a water barrel and a wine barrel, described here. The trick to solving the puzzle is that one need not make the calculations for each stage of the liquid ...
11
votes
5answers
400 views

How to prove that $\sum\limits_{i=0}^p (-1)^{p-i} {p \choose i} i^j$ is $0$ for $j < p$ and $p!$ for $j = p$

Let $p \in \mathbf{N}$. I don't know how to prove that $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^j=0 \textrm{ for } j \in \{0,\ldots,p-1\},$$ and $$\sum_{i=0}^p (-1)^{p-i} {p \choose i} i^p=p!$$ ...
4
votes
2answers
476 views

How can I compute $\sum\limits_{k = 1}^n \frac{1} {k + 1}\binom{n}{k} $?

This sum is difficult. How can I compute it, without using calculus? $$\sum_{k = 1}^n \frac1{k + 1}\binom{n}{k}$$ If someone can explain some technique to do it, I'd appreciate it. Or advice using ...
1
vote
1answer
82 views

A “fast” approach to compute $\sum_{i=0}^{n} \binom{19}{i} \times \binom{7}{n-i}$ [duplicate]

Possible Duplicate: How to find a closed formula for the given summation I am looking for a fast/best approach to compute $$\sum_{i=0}^{n} \binom{19}{i} \times \binom{7}{n-i}$$ For ...
1
vote
1answer
105 views

Combination Problem with a Variable

I have the following problem: $_xC_6$ = $_xC_4$ I expand both sides to: $$\frac{x!}{[(x-6)!]6!} = \frac{x!}{[(x-4)]!4!}$$ Next I multiply both sides by the denominator of the right-hand ...
5
votes
3answers
306 views

What does the notation $\binom{n}{i}$ mean?

What do the parentheses next to the summation involving the binomial coefficients mean? Like this: $$\sum _{i=0}^{n} \binom{n}{i}a^{(n-i)}b^i=\left(a+b\right)^n $$
2
votes
3answers
94 views

Further simplify this $\sum \limits_{i=1}^{k}{{k \choose i} \cdot 12^i \cdot 2^i}$

Could we further simplify this: $$ \sum_{i=1}^{k}{{k \choose i} \cdot 12^i \cdot 2^i}$$ or, at least, find a close upper bound?
14
votes
4answers
472 views

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
3
votes
1answer
137 views

Simplification of a sum

Any ideas on how to approximate and/or simplify this crazy-looking sum will be massively appreciated) $$\frac{1}{\mu}\sum_{j=0}^{\frac{\lambda}{2}} ...