How to find an explicit formula for a recursive function?

Question

Define $$ S_{n+1} = \frac{S_n^2+x}{2S_n}$$ and $S_1 = k$, where x,k > 0. find an explicit formula for $S_n$ in terms of n.

I don't even know where to begin. I tried using algebraic manipulation to rid of $S_{n+1}$ but nothing is working. Since x needs to disappear I'm thinking we should use ratios.

Answer 1

You can have the closed form

$$S_n = \sqrt {x}\coth \left( 2^{n-1}{ \coth^{-1}} \left( {\frac {k}{\sqrt {x}}} \right)\right) .$$

Added: If you want to find the limit without finding the closed form then you can advance as: assume $\lim_{n\to \infty } S_n = a $ then

$$ a=\frac{a^2+x}{2a} \implies a= \pm \sqrt{x}.$$