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Math Help - Let a>0 and let z1>0. Define zn+1:=for nN. show that (zn) converges and findthe limit

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    Let a>0 and let z1>0. Define zn+1:=for nN. show that (zn) converges and findthe limit

    Let a>0, and z1>0. Define zn+1= sqrt(a+zn), for n is a natural number. Show that {zn} converges and find the limit.
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    Re: Let a>0 and let z1>0. Define zn+1:=for nN. show that (zn) converges and findthe l

    Quote Originally Posted by TaTzRE66 View Post
    Let a>0, and z1>0. Define zn+1= sqrt(a+zn), for n is a natural number. Show that {zn} converges and find the limit.
    1. Prove that \{z_n\} is an increasing sequence.
    2. Prove that \{z_n\} has an upper bound.

    According to a well known theorem, \{z_n\} is convergent. If l is its limit then, l=\sqrt{a+l}. Solving this equation and taking into account that z_n>0 for all n, you'll find l=\frac{1+\sqrt{1+4a}}{2}.
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