Question

[lê thanh thủy]

Tính 1/1.3+1/3.5+1/5.7+...1/99.101

Answer 1

[soyeon_Tiểu bàng giải]

1/1.3 + 1/3.5 + 1/5.7 + ... + 1/99.101

= 1/2.(2/1.3 + 2/3.5 + 2/5.7 + ... + 2/99.101)

= 1/2.(1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + ... + 1/99 - 1/101)

= 1/2.(1 - 1/101)

= 1/2.100/101

= 50/101

Answer 2

[Edogawa Conan]

\(\text{Đặt : }A=\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+....+\frac{1}{99.101}\)

\(\Rightarrow2A=\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+....+\frac{2}{99.101}\)

\(\Rightarrow2A=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{101}\)

\(\Rightarrow2A=1-\frac{1}{101}\)

\(\Rightarrow A=\frac{100}{101}:2=\frac{50}{101}\)