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

Решите уравнение 10/x+10(1+9/х+9(1+8/х+8(...(1+1/х+1)...)))=11
- Предмет:
  Математика
- Автор:
  aylinwall
- 5 лет назад

Ответы 1

$\displaystyle \frac{10}{x}+10\left(1+\frac{9}{x}+9\left(1+\frac{8}{x}+8\left(\,\dotsc\,3\left(1+\frac{2}{x}+2\left(1+\frac{1}{x}+1ight)ight)ight)ight)ight)=11;$ $\displaystyle 10\left(\frac{1}{x}+1+9\left(\frac{1}{x}+1+8\left(\,\dotsc\,3\left(\frac{1}{x}+1+2\left(\frac{1}{x}+1+\frac{1}{x}+1ight)ight)ight)ight)ight)=11;$ $\displaystyle u=\frac{1}{x}+1;$ $\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+5\left(u+4\left(u+3\left(u+2\left(u+uight)ight)ight)ight)ight)ight)ight)ight)ight)=11;$ $\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+5\left(u+4\left(u+3\left(u+4uight)ight)ight)ight)ight)ight)ight)ight)=11;$ $\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+5\left(u+4\left(u+15uight)ight)ight)ight)ight)ight)ight)=11;$ $\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+5\left(u+64uight)ight)ight)ight)ight)ight)=11;$ $\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+6\left(u+325uight)ight)ight)ight)ight)=11;$ $\displaystyle 10\left(u+9\left(u+8\left(u+7\left(u+1956uight)ight)ight)ight)=11;$ $\displaystyle 10\left(u+9\left(u+8\left(u+13699uight)ight)ight)=11;$ $\displaystyle 10\left(u+9\left(u+109600uight)ight)=11;$ $\displaystyle 10\left(u+986409uight)=11;$ $\displaystyle 9864100u=11;$ $\displaystyle u=\frac{11}{9864100};$ $\displaystyle \frac{1}{x}+1=\frac{11}{9864100};$ $\displaystyle \frac{1}{x}=\frac{11-9864100}{9864100}=-\frac{9864089}{9864100};$ $\displaystyle x=\boxed{-\frac{9864100}{9864089}}\phantom{.}.$
- Автор:
  reubenriggs
- 5 лет назад
- 0
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