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Math Help - Another question on sequences

  1. #1
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    Another question on sequences

    I had another question:

    Let x1 be greater than or equal to 2. Define inductively x(n+1) as 1+square root(xn-1) for all n. Prove that (xn) is a decreasing sequence bounded below by 2. Find the limit.

    Thank you for any help. Fares.
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  2. #2
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    Quote Originally Posted by Fares23 View Post
    I had another question:

    Let x1 be greater than or equal to 2. Define inductively x(n+1) as 1+square root(xn-1) for all n. Prove that (xn) is a decreasing sequence bounded below by 2. Find the limit.

    Thank you for any help. Fares.
    The limit of I_n (nth iteration) is the same as I_{n+1} (n+1 th iteration).

    Thus,
    I_{n+1}=1+sqrt(I_n)
    Thus,
    If lim I_n= L then, lim I_{n+1}=L
    Thus,
    L=1+sqrt(L) (limit composition rule for sequnces)
    Solve for L.

    IMPORTANT. The fact that lim I_n exists was impervious to the proof. That I did not prove.

    But the idea is to show that it is an increasing bounded sequence. (Use Wierestrass-Bolzano Theorom)
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