Question

Find the value of $3+\frac{1}{6+\frac{1}{6+\frac{1}{6}+...}}$

Answer 1

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

$\Rightarrow x-3=\frac{1}{6+\frac{1}{6+\frac{1}{6}+...}}$ ...(i)

$\Rightarrow (x-3)=\frac{1}{6+\left(\frac{1}{6+\frac{1}{6+\frac{1}{6}+...}}\right)}$

$\Rightarrow (x-3)=\frac{1}{6+(x-3)}$ [From (i)]

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

$\Rightarrow (x-3)(x+3)=1$

$\Rightarrow x^2-9=1$

$\Rightarrow x^2=10$

$\therefore x=\sqrt{10}$