The golden number

An infinite nested radical is evaluated by noticing that the inner part, being infinite, equals the whole expression: from this a quadratic equation arises.

Example — The golden number

Let $S = \sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$.

The part that repeats is $S$ itself: $S = \sqrt{1+S}$.

Existence condition: $1+S\ge 0$ (satisfied for $S\ge 0$). Acceptability condition: $S\ge 0$.

Squaring: $S^2 = 1+S \implies S^2-S-1=0$. $S = \frac{1\pm\sqrt{5}}{2}$ Discarding the negative solution: $\boxed{S = \dfrac{1+\sqrt{5}}{2} = \varphi \approx 1{,}618}$ (the golden ratio).

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