Задача 68042 Найти произведение...

6308ea47f95bb26127f31346

31.12.2022 08:28:36

Найти произведение (1+1/5)(1+1/5^2)*...*(1+1/5^(16))

5f3ea7e3faf909182968ddd9

31.12.2022 09:19:24

★

Умножим и разделим на [m](1-\frac{1}{5})[/m]

Получим:

[m]\frac{(1-\frac{1}{5})(1+\frac{1}{5})(1+\frac{1}{5^2})(1+\frac{1}{5^4})\cdot ...\cdot(1+\frac{1}{5^{16}}) }{(1-\frac{1}{5})}=[/m]

Применяем формулу : [r][m] (a-b)(a+b)=a^2-b^2)[/m][/r]

[m]=\frac{(1-\frac{1}{5^2})(1+\frac{1}{5^2})(1+\frac{1}{5^4})\cdot ...\cdot(1+\frac{1}{5^{16}}) }{\frac{4}{5}}=[/m]


[m]=\frac{(1-\frac{1}{5^4})(1+\frac{1}{5^4})\cdot ...\cdot(1+\frac{1}{5^{16}}) }{\frac{4}{5}}=...[/m]


[m]=\frac{1-\frac{1}{5^{32}} }{\frac{4}{5}}=\frac{5(1-\frac{1}{5^{32}})}{4}=\frac{5-\frac{1}{5^{31}}}{4}[/m]