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

Question

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 $x$.

Answer 1

HINT:

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

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

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

Answer 2

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 $x$?