Question

[Ruby Sweety]

Tính T = \(\left(\frac{1}{2}-\frac{1}{3}\right)\left(\frac{1}{2}-\frac{1}{5}\right)\left(\frac{1}{2}-\frac{1}{7}\right).....\left(\frac{1}{2}-\frac{1}{99}\right)\)

Answer 1

[Hoàng Ninh]

\(T=\left(\frac{1}{2}-\frac{1}{3}\right)\left(\frac{1}{2}-\frac{1}{5}\right)\left(\frac{1}{2}-\frac{1}{7}\right).....\left(\frac{1}{2}-\frac{1}{99}\right)\)

\(T=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}-\frac{1}{7}-........-\frac{1}{99}\right)\)

\(T=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-..........-\frac{1}{99}\right)\)

\(T=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{99}\right)\)

\(T=\frac{1}{2}\left(\frac{99}{297}-\frac{3}{297}\right)\)

\(T=\frac{1}{2}.\frac{96}{297}\)

\(T=\frac{1}{2}.\frac{32}{99}\)

\(T=\frac{16}{99}\)

Answer 2

[Tran Le Khanh Linh]

\(T=\left(\frac{1}{2}-\frac{1}{3}\right)\left(\frac{1}{2}-\frac{1}{5}\right)\left(\frac{1}{2}-\frac{1}{7}\right)\cdot.....\cdot\left(\frac{1}{2}-\frac{1}{99}\right)\)

\(=\frac{1}{2\cdot3}\cdot\frac{3}{2\cdot5}\cdot\frac{5}{2\cdot7}\cdot.....\cdot\frac{97}{2\cdot99}\)

\(=\frac{1\cdot3\cdot5\cdot.....\cdot97}{2^{49}\left(3\cdot5\cdot7\cdot....\cdot99\right)}=\frac{1}{2^{49}\cdot99}\)