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Math Help - induction and recursion

  1. #1
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    induction and recursion

    give a recursive definaition of the sequence { {a_n}} , n=1,2,3,...
    a) a_n = 1+(-1)^n
    b) a_n = n(n+1)
    c) a_n = n^2
    help me how to define them
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  2. #2
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    (a)

    The sequence alternates between 0 and 2 starting with 0. The sum of two consecutive terms is 2. Hence

    a_1=0,\quad a_{n+1}=2-a_n


    (b)

    a_1=2,\quad a_{n+1}=(n+1)(n+2)=n(n+1)\frac{n+2}n=a_n\frac{n+2}  n


    (c)

    I leave this one to you.
    Last edited by proscientia; October 24th 2009 at 04:26 PM. Reason: Redo proofs
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  3. #3
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    c) a_n = a_n +2(n+1) -1 ,a_1=1 is that correct?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by proscientia View Post
    (a)

    The sequence alternates between 0 and 2 starting with 0. The sum of two consecutive terms is 2. Hence

    a_1=0,\quad a_{n+1}=2-a_n


    (b)

    a_1=2,\quad a_{n+1}=(n+1)(n+2)=n(n+1)\frac{n+2}n=a_n\frac{n+2}  n


    (c)

    I leave this one to you.
    The idea is to remove all reference to n except in subscripts from the expression. That is a_n should depend on a_{n-1},\ a_{n-2}, \ ... and not explicitly on n

    CB
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by zpwnchen View Post
    help me how to define them
    See here

    CB
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