Question

Tính hợp lý

a) A = \(\left(1-\frac{1}{1.2}\right)+\left(1-\frac{1}{2.3}\right)+...+\left(1-\frac{1}{2016.2017}\right)\)

 

Answer 1

Ta có : \(A=\left(1-\frac{1}{1.2}\right)+\left(1-\frac{1}{2.3}\right)+.......+\left(1-\frac{1}{2016.2017}\right)\)

\(\Rightarrow A=\left(1+1+1+......+1\right)-\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+......+\frac{1}{2016.2017}\right)\)

\(\Rightarrow A=2016-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+......+\frac{1}{2016}-\frac{1}{2017}\right)\)

\(\Rightarrow A=2016-\left(1-\frac{1}{2017}\right)\)

\(\Rightarrow A=2016-\frac{2016}{2017}=2015\frac{1}{2017}\)