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I am trying to solve this problem but nothing is thought on my mind. Please any one help me to solve this problem

$\sqrt{x + \sqrt {x + \sqrt{ x + \cdots } } } = 5$ then find the value of $x$.

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marked as duplicate by MJD, Sami Ben Romdhane, Xoff, Davide Giraudo, egreg Jan 15 '14 at 15:42

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

look here: math.stackexchange.com/q/638048/119592 I believe that is similar –  Bernd Jan 15 '14 at 14:08

2 Answers 2


$$\text{If }\sqrt{\underbrace{x+\sqrt{\underbrace{x+\sqrt{x+\cdots}}}}}=y,$$

As $\infty-1=\infty,$ the terms under the two braces are same i.e.,

$$ \sqrt{x+y}=y$$

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You are given $$\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}=5\tag{1}$$ Squaring both sides gives you $$x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}=25\tag{2}$$ Do you see how you can use these two equations to isolate $x$?

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