Problem 33. Nested Square Roots
Suggested question by https://twitter.com/sina_yari13:
Find the value of x in the following expression: \[x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}} = 625.\]
Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1699714621899481219
If we take the square roots of both sides, we have:
\[\sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + ...}}}} = 25.\]
The above expression is equivalent to the part under the sequare root of the original expression. Thus, we can replace it with \(25\) to get the following equation:
\[x + 25 = 625\]
Which implies \(x = 600\).