# binomial expansion

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• Nov 8th 2009, 06:40 PM
purebladeknight
binomial expansion
i just need verification, tell me if i am correct or not. if im incorrect please correct me.

1.By using the binomial expansion, show that:

$(q+p)^n - (q-p)^n = 2 (nC1) q^(n-1)p + 2 (nC3)q^(n-3)p^3 +.....$

i expanded the first two terms of (q+p)^n and -(q-p)^ n, then i added them together to get the exact statement above. is that fine or do i need a vigorous show to n terms?

2. what is the last term of the expansion if n is odd

if n is odd then the last two terms of (q+p)^n and -(q-p)^n are added together giving $2(nCk) p^k$

3. what is the last term of the expansion if n is even

if n is even then the last two terms of $(q+p)^n$ and $-(q-p)^n$ cancel each other leaving the second last term [tex] 2 (nC(k-1)) q. p^(k-1)
• Nov 9th 2009, 04:07 PM
kjchauhan
$(q+p)^n - (q-p)^n=$ $\sum_{k=0}^{n} \binom{n}{k}\ q^k p^(k-n)$ $-\sum_{k=0}^{n} (-1)^n\binom{n}{k}\ q^k p^(k-n)$

Now expand the sum..(Happy)