Question

 

 

Tính  giá trị biểu thức:  \(\frac{1}{1\cdot3}\)+\(\frac{1}{3\cdot5}\)+\(\frac{1}{5\cdot7}\)+.....+\(\frac{1}{99\cdot101}\)

Answer 1

Đặt biểu thức trên là A = 1/1.3 + 1/3.5 + 1/5.7 +...+ 1/99.101

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

2A= 2/1.3 + 2/3.5 + 2/ 5.7 +... 2/99.101

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

2A= 1-1/101

2A= 100/101

A= 100/101 . 1/2

A= 50/101

Answer 2

\(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{99.101}\)

\(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{101}\)

\(\Rightarrow1-\frac{1}{101}=\frac{100}{101}\)

Answer 3

Ta có : \(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+.......+\frac{1}{99.101}\)

\(=\frac{1}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+......+\frac{2}{99.101}\right)\)

\(=\frac{1}{2}\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+.......+\frac{1}{99}-\frac{1}{101}\right)\)

\(=\frac{1}{2}\left(1-\frac{1}{101}\right)\)

\(=\frac{1}{2}.\frac{100}{101}=\frac{50}{101}\)