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Math Help - Induction proof to prove recursive formula is equal to explicit formula

  1. #1
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    Induction proof to prove recursive formula is equal to explicit formula

    I'm really stuck on these, any help is appreciated. In each of the following a sequence is defined recursively. Guess an explicit formula for the sequence, then use mathematical induction to prove the correctness of your formula. Sorry I had some font sizing issues. If you can just explain the first one, then I can probably figure out the other two. Thanks.


    (a) csub(k) = 3csub(k- 1) + 1, for all integers k >= 2
    c1 = 1



    (b) gsub(k) = (gsub(k -1) ) / (gsub(k- 1) + 2) , for all integers k >= 2
    g1 = 1



    (c) psub(k) = psub(k -1) + 2 3^k, for all integers k >= 2
    p1 = 2

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  2. #2
    Senior Member MacstersUndead's Avatar
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    Re: Induction proof to prove recursive formula is equal to explicit formula

    I think I might be spoiling you because in this format it's hard to read and not likely to get accepted.

    (a) ck = 3ck-1 + 1 for all integers k \geq 2 (INDUCTION STATEMENT P)
    c1 = 1

    However,
    ck-1 = 3ck-2 + 1, so
    ck = 3 (3ck-2 + 1) + 1 = (3^2)ck-2 + 2 (MODIFIED INDUCTION STATEMENT P)

    ((Proceed inductively by replacing ck-m term until you reach c1 = 1 ))

    = 3^(k-1)c1 + k
    = 3^(k-1) + k

    Prove this statement by general induction if needed.
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