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Thread: induction and recursion

  1. #1
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    induction and recursion

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

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

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


    (b)

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


    (c)

    I leave this one to you.
    Last edited by proscientia; Oct 24th 2009 at 03:26 PM. Reason: Redo proofs
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  3. #3
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    c) $\displaystyle a_n = a_n +2(n+1) -1 ,a_1=1$ is that correct?
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  4. #4
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    Quote Originally Posted by proscientia View Post
    (a)

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

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


    (b)

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


    (c)

    I leave this one to you.
    The idea is to remove all reference to $\displaystyle n$ except in subscripts from the expression. That is $\displaystyle a_n$ should depend on $\displaystyle 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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