last terms of expansion

**purebladeknight** · November 8th 2009, 07:06 PM

$(q+p)^n - (q-p)^n = 2 ({n}C1) q^{n-1}p + 2 ({n}C3)q^{n-3}p^3 +.....$

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

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

how do i find last terms?

**Grandad** · November 8th 2009, 11:03 PM

Hello purebladeknight

Originally Posted by purebladeknight

$(q+p)^n - (q-p)^n = 2 ({n}C1) q^{n-1}p + 2 ({n}C3)q^{n-3}p^3 +.....$

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

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

how do i find last terms?

You'll have realised, of course, that the 1st, 3rd, ... terms (the odd terms) in the expansions of $(q+p)^n$ and $(q-p)^n$ have been eliminated by subtraction, leaving us with the even terms, which are then added together to give the factor of $2$ in each term in the final expression.

Then notice that in the expansion of $(q+p)^n$ there are $n+1$ terms altogether, the last two terms being

$...+\underbrace{nqp^{n-1}}_{n^{th}\text{ term}}\quad + \underbrace{p^n}_{(n+1)^{th}\text{ term}}$

So if $n$ is odd, $n+1$ is even, and the final term will be $2p^n$ .

And if $n$ is even, $n+1$ is odd, and the final term will be $2nqp^{n-1}$ .

Grandad

**purebladeknight** · November 9th 2009, 12:18 AM

Originally Posted by Grandad

Hello purebladeknightYou'll have realised, of course, that the 1st, 3rd, ... terms (the odd terms) in the expansions of $(q+p)^n$ and $(q-p)^n$ have been eliminated by subtraction, leaving us with the even terms, which are then added together to give the factor of $2$ in each term in the final expression.

Then notice that in the expansion of $(q+p)^n$ there are $n+1$ terms altogether, the last two terms being

$...+\underbrace{nqp^{n-1}}_{n^{th}\text{ term}}\quad + \underbrace{p^n}_{(n+1)^{th}\text{ term}}$

So if $n$ is odd, $n+1$ is even, and the final term will be $2p^n$ .

And if $n$ is even, $n+1$ is odd, and the final term will be $2nqp^{n-1}$ .

Grandad

perfect! thank you Grandad, an excellent explanation.

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