Tính giá trị biểu thức: A=(1-1/1.2) + (1-1/2.3)+...+(1-1/2015.2016)

* Hướng dẫn giải

$\begin{array}{l}A=\left(1-\frac{1}{1.2}\right)+\left(1-\frac{1}{2.3}\right)+\mathrm{...}+\left(1-\frac{1}{2015.2016}\right)\\ =1+1+1+\mathrm{...1}-\left(\frac{1}{1.2}+\frac{1}{2.3}+\mathrm{...}+\frac{1}{2015.2016}\right)\\ =2015-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\mathrm{...}+\frac{1}{2015}-\frac{1}{2016}\right)\\ =2015-\left(1-\frac{1}{2016}\right)\\ =2015-1+\frac{1}{2016}\\ =2014+\frac{1}{2016}\\ =\frac{4060225}{2016}\end{array}$