Nested radical with sums

The same trick works when a term appears in front of the radical: you isolate the radical, square it and obtain a quadratic equation.

Example — Nested radical with sums

$G = 2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$. Then $G = 2+\sqrt{G}$, that is $(G-2)^2 = G$: $G^2-5G+4=0 \implies (G-1)(G-4)=0 \implies \boxed{G=4}$ (discarding $G=1$ because $G \ge 2$).

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Topics: Quadratic equations
Concepts: Quadratic equation · Nested radical
Skills: Solving equations