Jun 14, 2021 - 3:34

An infinite series problem.

By Vamshi Jandhyala in mathematics

June 14, 2021

Problem

Solution

Let

$$ x = \sqrt{n + \sqrt{n + \sqrt{n + \cdots}}} $$

We have

$$ \begin{aligned} x &= \sqrt{x + n} \\
\implies x^2 &= n + x \\
\implies x &= \frac{1 + \sqrt{4n+1}}{2} \text{ as $x$ is positive} \end{aligned} $$

Therefore,

$$ \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}} = \frac{1 + \sqrt{5}}{2} $$