Nested Radical

Let $f(x) = x+1$ . Then

$\begin{aligned} f(x) & = x+1 = \sqrt{(x+1)^2} = \sqrt{1+x^2+2x} = \sqrt{1+x\color{#3D99F6}(x+2)} & \small \color{#3D99F6} \text{Note that } x+2 = x+1 +1 = f(x+1) \\ & = \sqrt{1+xf(x+1)} \\ & = \sqrt{1+x\sqrt{1+(x+1)f(x+2)}} \\ & = \sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)f(x+3)}}} \\ \implies x + 1 & = \sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+(x+3)\sqrt{1+\cdots}}}}} & \small \color{#3D99F6} \text{Putting }x=2 \\ \implies \boxed 3 & = \sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\cdots}}}}} \end{aligned}$

1. In Brilliant's own wiki: Nested Functions . There is a link in the wiki page to a previous posting.

2. Previously posted: Nested Radicals

3. See Chew-Seong Cheong 's adjoining excellent solution.