Let a>0, and z_{1}>0. Define z_{n+1}= sqrt(a+z_{n}), for n is a natural number. Show that {z_{n}} converges and find the limit.

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- Sep 19th 2012, 10:32 PMTaTzRE66Let a>0 and let z1>0. Define zn+1:=for nN. show that (zn) converges and findthe limit
Let a>0, and z

_{1}>0. Define z_{n+1}= sqrt(a+z_{n}), for n is a natural number. Show that {z_{n}} converges and find the limit. - Sep 19th 2012, 10:44 PMFernandoRevillaRe: Let a>0 and let z1>0. Define zn+1:=for nN. show that (zn) converges and findthe l
1. Prove that $\displaystyle \{z_n\}$ is an increasing sequence.

2. Prove that $\displaystyle \{z_n\}$ has an upper bound.

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