Simplifying $\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt {5 +\cdots}}}}$

Solution 1:

We should really make the problem precise, and prove convergence. But this is the GRE, we manipulate. Let $x$ be the number. Then $x^2-5=2x$. Our number is the positive root of the quadratic.

Solution 2:

Let $x = 2\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt{...}}}}$. Then (if this converges) $x = 2\sqrt{5+x}$. Solving, $x = 2(1+\sqrt6)$, so the answer to your original question is $1+\sqrt{6}$

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