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.
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 $\displaystyle l=\frac{1+\sqrt{1+4a}}{2}.$