Question

Show that $P+\frac{2}{1+\frac{1}{P+\frac{1}{1+P...}}}=\sqrt{P^2+4P}$

Answer 1

$P+\frac{2}{1+\frac{1}{P+\frac{1}{1+P...}}}=x$ [say]

$(x-p)=\frac{2}{1+\frac{1}{(P+x)/2}}$

$\Rightarrow (x-p)=\frac{2(P+x)}{2+P+x}$

$\Rightarrow (x-p)(p+x+2)=2P+2x$

$\Rightarrow Px+x^2+2x-P^2-Px-2P=2P+2x$

$\Rightarrow x^2=P^2+4P$

$\Rightarrow x=\sqrt{P^2+4P}$