Non linear Recursion demonstration

Question

I wrote a code to find the $ \lim_{n \to ∞} F_n \ $ where $a > 0$ and $F_n$ is defined as: $$F_0=x_0$$ $$F_{n+1}=0.5(F_n+\frac{a}{F_n})$$

By analysing the output of the code and trying different values for $a$, I found out that the $ \lim_{n \to ∞} F_n = a^{0.5}\ $

I want to understand why. Can someone help me with a small demonstration. Also, I would appreciate if someone give me the correct term to research about this particular type of recursion for my further readings.

Here is my code:

Answer 1

Rewrite $$F_{n+1}=\frac 12 \left(F_n+\frac{a}{F_n} \right)=F_n-\frac 12 \left(F_n-\frac{a}{F_n} \right)=F_n-\frac{F_n^2-a}{2 F_n }$$ This is also Newton's method iterative scheme for solving $F^2=a$ starting with $F_0$.