Binomial theorem Definition and 138 Threads

Homework Statement Use the binomial theorem to find the coefficient of ##x^6y^3## in ##(3x-2y)^9##. Homework Equations ##1+9+36+84+126+126+84+36+9+1## (I used two lines for the lenght) ##1(3x)^9(-2y)^0+9(3x)^8(-2y)^1+36(3x)^7(-2y)^2+84(3x)^6(-2y)^3+126(3x)^5(-2y)^4##...

Homework Statement Use the binomial theorem to find the coefficient of ##x^8y^5## in ##(x+y)^{13}##. Homework Equations We know 13 - 5 = 8 , so we have ##\binom{n}{5}x^{n-5}y^5 = \binom{13}{5}x^8y^5## ##\binom{13}{5} = \frac{13 \cdot 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8!}{5!8!} = \frac{13...

1.Find n, if the term 11 coefficient it is 6 time the term 10 coefficient in 2.(6x^7 + 5x^(-4))^n 3

In page 11 of http://math.arizona.edu/~zakharov/BesselFunctions.pdf, I am trying to follow the derivation using binomial theorem to get this step: (e^{j\theta}-e^{-j\theta})^{n+2k}≈\frac{(n+2K)!}{k!(n+k)!}(e^{j\theta})^{n+k}(-e^{-j\theta})^kIf you read the paragraph right above this equation...

Here is the question: Here is a link to the question: What are the last three and four terms of (a + b)^n? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Will someone be kind enough to check my proof (attached) of the following (also attached) theorem?Homework Equations The Attempt at a Solution Oh, and as you might notice, I was beginning to run out of paper, but the binomial coefficients in the bracketed terms obviously...

Guess what? I just got my new calculus book last week! ^^ The book opens with the definition of the real numbers by Dedekind and goes to prove properties of this numbering system such as The supremum axiom and others. At the end of the chapter are about 30 exercises without their solutions...

Homework Statement Define (n k) = n!/k!(n-k)! for k=0,1,...,n. Part (b) Show that (n k) + (n k-1) = (n+1 k) for k=1,2,...n. Part (c) Prove the binomial theorem using mathematical induction and part (b). Homework Equations The Attempt at a Solution I'm wasn't able to...

Homework Statement Prove that n^n > 2^n * n! when n > 6 using the Binomial theorem. I just proved the Binomial theorem using induction which was not that difficult but in applying what I learned through it's proof I am having difficulty. Homework Equations Binomial theorem = (x+y)^n =...

Homework Statement (√2 + 1)6 = I + f Where I is the sum of integer part of the expansion of (√2 + 1)6 and f is sum of the fraction part in (√2 + 1)6. Homework Equations (x+1)n = nC0 xn + nC1 xn-1 + nC2 xn-2 + ... + nCn nCn = nC0 = 1 The Attempt at a Solution I expanded...

I have been teaching myself analysis with baby rudin. I have just started chapter three in the past week or so and one thing I am having trouble with is the proofs which use the binomial theorem and various identities derived from it. Rudin pretty much assumes this material as prerequisite and...

Homework Statement Prove the binomial theorem by induction. The attempt at a solution http://desmond.imageshack.us/Himg35/scaled.php?server=35&filename=sumu.png&res=landing Hi, running into trouble with this proof and google hasn't helped me. I don't understand the jump here, and as...

Homework Statement What am I supposed to do with the 3 over 2 in the parentheses? It can be divide and it can be take the factorial. So what do I do with it?

Homework Statement Find the first four terms in the expansion of \left(1-3x\right)^{3/2}. By substituting in a suitable value of x, find an approximation to 97^{1/2}. Homework Equations The Attempt at a Solution I used the binomial expansion formula to work the answer and it is 1-...

I was trying to make sense of the equation attached below which was on the wikipedia site. However I'm not entirely sure how to make use of the "n choose 0" , "n choose 1", etc. statements that in front of each term in of the expansion. I roughly know how the expansion should look...

Homework Statement The method of Binomial expansion is useful because you can avoid expanding large expressions: Q: Find the term indepedent of x in the expansion of (2+x)[2x+(1/x)]5 The attempt at a solution: "For this to produce a term independent of x, the expansion of [2x+(1/x)]5 must...

Homework Statement Find an approximation of (0.99)5 using the first three terms of its expansion. 2. The attempt at a solution To get to the binomial theorem I divided 0.99 into (0.99)5 = (1-0.01)5 = {1+(-0.01)}5 Then, T1 = 5C0(1)5 = 1 x 1=1 T2 = 5C1(1)5-1(-0.01)1 = 5x1x...

Use Binomial Theorem and appropriate inequalities to prove! Homework Statement Use Binomial Theorem and appropriate inequalities to prove 0<(1+1/n)^n<3 Homework Equations The Attempt at a Solution So I started by.. \sum ^{n}_{k=0} (n!/(n-k)! k!) a^{n-k}b^{k} = n!/(n-k)!k! (1)^{n-k}...

Homework Statement (1+n)^n≥ 5/2* n^n- 1/2* n^(n-1) for n≥2 Homework Equations i know I have to use this formula (1+x)^n=1+nx/1!+(n(n-1) x^2)/2!+⋯ The Attempt at a Solution And you take x=n from my original inequality but after that I have no clue (1+n)^n=1+n/1! n+(n(n-1) n^2)/2!+⋯ but it...

Homework Statement I am doing a poof and I need to use the binomial theorem. However is the following a correct way to rewrite it? (a+b)^n\ =\ {n \choose 0}a^{n} + \sum_{k=1}^{n}{n \choose k}\ a^{n-k}\ b^{k} Homework Equations (a+b)^n\ =\ \sum_{k=0}^{n}{n \choose k}\ a^{n-k}\ b^{k}...

Homework Statement Use the binomial theorem to rpove that for n a positive integer we have: (1 + 1/n)^n = 1 + sum(k=1 to n) [1/k! product(r=0 to k-1) (1 - r/n)] The Attempt at a Solution (1 + 1/n)^n = 1 + sum(k=1 to n) (n choose r) 1^n-k (1/n)^k, where (n choose r) = n!/r!(n - r)...

Homework Statement Show that if the greatest term in the expansion of (1+x)2n is also the greatest coefficient, then x lies between n/n+1 and n+1/n. Homework Equations No idea. The Attempt at a Solution Don't know where to start.

Homework Statement Let a be a fixed positive rational number. Choose (and fix) a natural number M>a. Use (a^n)/(n!)\leq(a^M/(M!))(a/M)^(n-M) to show that, given e>0, there exists an N\inN such that for all n\geqN, (a^n)/n! < e. Homework Equations The Attempt at a Solution In a...

Homework Statement Let a be a fixed positive rational number. Choose(and fix) a naural number M > a. a) For any n\inN with n\geqM, show that (a^n)/(n!)\leq((a/M)^(n-M))*(a^M)/(M!) b)Use the previous prblem to show that, given e > 0, there exists an N\inN such that for all n\geqN, (a^n)/(n!)...

Homework Statement Use the above to prove that given a rational number a > 1 and A any other rational number, there exists b ε N such that ab > A. Homework Equations The above refers to the proving, by use of both induction and binomial theorem, that (1+a)n ≥ 1+na. Binomial Theorem: (i=0 to...

Hi, I am trying to understand the binomial theorem, and would appreciate any insight or pointers. To make notation simpler I'll call the binomial coefficient f(n,k). I understand the combinatorial argument that f(n,k) = f(n-1, k-1) + f(n-1, k). This is, to my understanding, a two...

Homework Statement Homework Equations The Attempt at a Solution I am really stuck, I have no clue how to even begin. For part B I tried changing the RHS to factorials but I was left at a dead end there.

Homework Statement Consider an ideal gas of N identical particles in a volume V, and a subvolume v. The chance a molecule is in inside the subvolume is P = v/V. a) What is the chance the subvolume contains n particles? b) Use the binomial theorem (p + q)^N = \sum_{n = 0}^N p^n q^{N-n}...

Homework Statement Use Newton's Binomial Theorem to estimate integral of (1+x^4)^(1/2) from 0 to 1/2 to within one part in 1000, (error>1/1000) Homework Equations I used the Binomial Series expansion, so (a+b)^n = a^n +na^(n-1)b + (n(n-1))/2! (etc The Attempt at a Solution I...

Where did the binomial theorem come from...?

Homework Statement If n \in N, then 11n+2 + 122n+1 is divisible by:- a)113 b)123 c)133 Homework Equations The Attempt at a Solution I did it by substituting different values of n and divided by each of the option. Answer came out to be 133. But I want to do it step by step...

Homework Statement Find the coefficient of http://webwork2.math.utah.edu/webwork2_files/tmp/equations/73/3e29a3b979c709dbb6c609c5a6ce891.png in the expansion of [PLAIN][PLAIN]http://webwork2.math.utah.edu/webwork2_files/tmp/equations/63/dcb58790e8122dce61b830977294091.png Homework Equations...

Using summation((\stackrel{n}{k})xkyn-k) = (x+y)n, I let x = y = 1. This should then result in summation((\stackrel{n}{k})*1*1) = (1 + 1)n = 2n. Expanding the summation, I get (\stackrel{n}{0}) + (\stackrel{n}{1}) + ... +(\stackrel{n}{n}) = 2n. Solving this results in...

Prove for all n\inN 2n= (\stackrel{n}{0})+(\stackrel{n}{1})+...+(\stackrel{n}{n}) So I used mathematical induction base case: n=0 so 20=1 and (\stackrel{0}{0})=1 induction step: Let n\inN be given, assume as induction hypothesis that 2n=...

Hi guys, I'm Filip and as a 11th grade student I have a question about one mathematical problem. It says: If the coefficient of xk in the expansion of (3+2x-x2 )*(1+x)34 is zero. Find the value of k. I know it's something related with binomial theorem, but I don't really know how to start. Thank...

Homework Statement To Prove: (nC0)(mC0) + (nC1)(mC1) + ... + (nCm)(mCm) = (n+m C m) where nC0 = n choose 0 and so on. Homework Equations The Attempt at a Solution Tried expanding the whole thing using factorials - but didn't work. Any hints would be really welcome!

In my book, it says that the Binomial Series is \sum_{n=0}^{\infty }\binom{n}{r} x^n Where \binom{n}{r} = \frac{n(n-1)...(n-r+1)}{n!} for r\geq1 and \binom{n}{0} = 1 Now here is where it got to be, I know that the \binom{n}{r} = \frac{n(n-1)...(n-r+1)}{n!} were derived through the...

Homework Statement (here, (n,k) reads n choose k) prove that (n,0) - (n, 1) + ... + (-1)n(n,n) = 0 Homework Equations binomial theorem The Attempt at a Solution so this proof is relatively straightforward when n is odd. it's just matching up terms and having them cancel each other...

i was told the binomial theorem is as follows: (1-x)^n = 1-nx+ (n(n-1)/2!)x^2 - (n(n-2)/2!)x^3 ... not sure if this is right could some one clear this doubt for me any help is appreciated was told this in a physics class

Homework Statement (n¦0)-(n¦1)+(n¦2)-. . . ± (n¦n)=0 that reads n choose zero and so on Homework Equations Prove this using the binomial theorem The Attempt at a Solution I really have no idea where to start. Any help would be greatly appreciated thanks

Homework Statement Calculate \sqrt{1/20} using the extended binomial theormem. (a precision of k=4 is enough) The Attempt at a Solution \sqrt{1/20}= (1 + (-19/20) )^{1/2}= \sum( choose (1/2,k)*(-19/20)^k) = 1- 1/2*19/20-1/8*361/400+1/16*6589/8000 = 0.72... is wrong. Homework...

I have this question and its a combination of the binomial theorem and nilpotent elements within a ring. Suppose the following, am=bn=0. Is it necessarily true that (a+b)m+n is nilpotent. For this question I did the following: \sumi=0m+n\binom{m+n}{i}am+n-ibi If i=m, then a=0...

Homework Statement prove that (\stackrel{2n}{n}) is even when n \geq1 Homework Equations as a hint they gave me this identity: \stackrel{n}{k}= (n/k)(\stackrel{n-1}{k-1}) The Attempt at a Solution by using that identity i got: (\stackrel{2n}{n}) = (2n/n)...

Homework Statement \mbox{Prove or give a counterexample: If p is a prime integer, then for all integers x and y, } (x+p)^p \equiv_p x^p+y^p. Homework Equations \equiv_p \mbox{just means (mod p). Can you please check and see if this proof is well-formed?} The Attempt at a Solution...

Homework Statement Homework Equations Formula => C(n,r) or nCr =n!/r!(n-r)! & the basic Binomial Theorem formula. *Answer mentioned in book = nx The Attempt at a Solution The LHS should be (x+y)n & the given question is its expansion only if that 'r' is not multiplied in the question. I...

Hello, All we know the Binomial Theorm which may be stated mathematically as: \left(x+y\right)^n=\sum_{k=0}^n{n\choose k}y^k\,x^{n-k} Now suppose that we have the following mathematical expression: \sum_{k=0}^{n}{n\choose k}\,(-1)^k if we substitute x=1 and y=-1 in the first...

use the binomial theorem to determine the coefficient of x^33 in the expansion of (\frac{1}{4}-2x^3)^17 ive played around with it and come up with 33^C_17 as a coefficient.am i right in saying that is all the question asks Homework Equations The Attempt at a Solution

Using the Binomial Theorem and the definition of the derivate of a function f(x) as f'(x)= lim as h tends to 0 ((f(x+h)-f(x))/h) Prove that if f(x)=x^n then f'(x)=nx^(n-1) I'm confused as to how to exactly incorporate the nCr "n choose r" into this interpretation of the...

I am asked to prove that \sum ^n _{k=0} (C^n_k)^2 = C^{2 n}_n Where C^n_k signifies "n choose k" I am told the hint to use the binomial theorem and to calculate the coefficient of x^n in the product (1+x)^n (1+x)^n = (1+x)^{2n} the Binomial theorem is given by (x+y)^n = \sum_{k=0}...

Rudin's proof of lim n-> inf (p^(1/n)) = 1 1+n*x_n <= (1 + x_n)^n = o I don't see it from the binomial theorem, which is what he says that is from. He also does things with the binomial theorem like: (1+x_n)^n >= ((n(n-1)) / 2) *x_n^2 I'm not sure what he did to get these two...