How to find the value of $\sqrt{20+\sqrt{20+\sqrt{20 + \cdots}}}$

Question

Generally, I know how to calculate the square roots or cube roots, but I am confused in this question, not knowing how to do this:

$$\sqrt{20+\sqrt{20+\sqrt{20 + \cdots}}}$$

Note: Answer given in the key book is $5$. Not allowed to use a calculator.

Answer 1

HINT:

Let $\displaystyle S=\sqrt{20+\sqrt{20+\sqrt{20+\cdots}}}$ which is definitely $>0$

$\displaystyle\implies S^2=20+S\iff S^2-S-20=0$

But we need to show the convergence of the sum

Answer 2

Denote the corresponding value by $x$, then it satisfies the relation $$x=\sqrt{20+x},$$ with the only positive solution $x=5$.