Question

Show that $3+\frac{1}{1+}\frac{1}{6+}\frac{1}{1+}\frac{1}{6+}....=3(1+\frac{1}{3+}\frac{1}{2+}\frac{1}{3+}\frac{1}{2+}.....)$.

Answer 1

Let $x=3+\frac{1}{1+}\frac{1}{6+}\frac{1}{1+}\frac{1}{6+}......$

$\Rightarrow x-3=\frac{1}{1+\frac{1}{6+x-3}}=\frac{x+3}{x+4}$

$\Rightarrow x^2+x-12=x+3$

$\Rightarrow x=\sqrt{15}$ taking positive root

Let $y=1+\frac{1}{3+}\frac{1}{2+}\frac{1}{3+}\frac{1}{2+}......$

$\Rightarrow y-1=\frac{1}{3+\frac{1}{2+y-1}}=\frac{y+1}{3y+4}$

$\Rightarrow 3y^2+y-4=y+1$

$\Rightarrow 3y^2=5$

$\Rightarrow y=\sqrt{\frac{5}{3}}$

$x=3y$

Hence, proved.