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Math Help - last terms of expansion

  1. #1
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    last terms of expansion

     (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?
    Last edited by purebladeknight; November 8th 2009 at 10:17 PM.
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  2. #2
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    Hello purebladeknight
    Quote Originally Posted by purebladeknight View Post
     (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
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  3. #3
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    Quote Originally Posted by Grandad View Post
    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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