Question

Find the value of $3+\frac{1}{1+}\frac{1}{2+}\frac{1}{3+}\frac{1}{1+}\frac{1}{2+}\frac{1}{3+}....$.

Answer 1

Given, $3+\frac{1}{1+}\frac{1}{2+}\frac{1}{3+}\frac{1}{1+}\frac{1}{2+}\frac{1}{3+}.....$

$\Rightarrow x=3+\frac{1}{1+\frac{1}{2+\frac{1}{3+(x-3)}}}=3+\frac{1}{1+\frac{x}{2x+1}}$

$\Rightarrow x=3+\frac{2x+1}{3x+1}$

$\Rightarrow x=\frac{11x+4}{3x+1}$

$\Rightarrow 3x^2-10x-4=0$

$\Rightarrow x=\frac{10\pm \sqrt{148}}{6}$

Thus $x=$ positive root

Hence, $x=\frac{10\pm \sqrt{148}}{6}=\frac{5+\sqrt{37}}{3}$