Question

Решите уравнение 10/x+10(1+9/х+9(1+8/х+8(...(1+1/х+1)...)))=11

Answer 1

$\displaystyle \frac{10}{x}+10\left(1+\frac{9}{x}+9\left(1+\frac{8}{x}+8\left(\,\dotsc\,3\left(1+\frac{2}{x}+2\left(1+\frac{1}{x}+1\right)\right)\right)\right)\right)=11;$

$\displaystyle 10\left(\frac{1}{x}+1+9\left(\frac{1}{x}+1+8\left(\,\dotsc\,3\left(\frac{1}{x}+1+2\left(\frac{1}{x}+1+\frac{1}{x}+1\right)\right)\right)\right)\right)=11;$

$\displaystyle u=\frac{1}{x}+1;$

$\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+5\left(u+4\left(u+3\left(u+2\left(u+u\right)\right)\right)\right)\right)\right)\right)\right)\right)=11;$
$\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+5\left(u+4\left(u+3\left(u+4u\right)\right)\right)\right)\right)\right)\right)\right)=11;$
$\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+5\left(u+4\left(u+15u\right)\right)\right)\right)\right)\right)\right)=11;$
$\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+5\left(u+64u\right)\right)\right)\right)\right)\right)=11;$
$\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+325u\right)\right)\right)\right)\right)=11;$
$\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+1956u\right)\right)\right)\right)=11;$
$\displaystyle 10\left(u+9\left(u+8\left(u+13699u\right)\right)\right)=11;$
$\displaystyle 10\left(u+9\left(u+109600u\right)\right)=11;$
$\displaystyle 10\left(u+986409u\right)=11;$
$\displaystyle 9864100u=11;$

$\displaystyle u=\frac{11}{9864100};$

$\displaystyle \frac{1}{x}+1=\frac{11}{9864100};$

$\displaystyle \frac{1}{x}=\frac{11-9864100}{9864100}=-\frac{9864089}{9864100};$

$\displaystyle x=\boxed{-\frac{9864100}{9864089}}\phantom{.}.$