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Math Help - binomial expansion

  1. #1
    Junior Member
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    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)
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  2. #2
    Member kjchauhan's Avatar
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    (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..
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