# Jun 14, 2021 - 3:34

#### An infinite series problem.

By Vamshi Jandhyala in mathematics

June 14, 2021

## Problem

— 級数bot (@infseriesbot) June 14, 2021

## 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,

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