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Math Help - Sequence Proof

  1. #1
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    Exclamation Sequence Proof

    i need to prove the following:

    let:
    a1 = 3
    an+1 = (1/3)(an + 12)

    for all n in N. show that the sequence an converges, and then find the limit

    Thanks!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by dlee426 View Post
    i need to prove the following:

    let:
    a1 = 3
    an+1 = (1/3)(an + 12)

    for all n in N. show that the sequence an converges, and then find the limit

    Thanks!
    You need to first show that the sequence is bounded from above [i.e. prove by induction that a_k<M\,(M\in\mathbb{R})] and show that it is monotone [prove by induction that s_{k}<s_{k+1}]. Once you've done that, note that as n\to\infty, a_{n+1}\to a,\,a_{n}\to a. Thus a=\tfrac{1}{3}\left(a+12\right)\implies a=\dots

    Can you take it from here?
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