Question

Решите уравнение:
$\frac{ 1+\frac{1+ \frac{...}{5} }{5}}{5}+1=x$

Answer 1

$\frac{1+\frac{1+\frac{\cdots}{5}}{5}}{5} = \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \cdots, \\ b_1= \frac{1}{5} , q = \frac{1}{5}, \\ S= \frac{b_1}{1-q} = \frac{\frac{1}{5}}{1-\frac{1}{5}} = \frac{1}{4} , \\ \frac{1+\frac{1+\frac{\cdots}{5}}{5}}{5}+1=x, \\ x=1\frac{1}{4}.$