Question

\(D=\left(1-\frac{1}{1.2}\right)+\left(1-\frac{1}{2.3}\right)+...+\left(1-\frac{1}{2015-2016}\right)\)

Ai giai giup minh voi

 

Answer 1

\(\left(1-\frac{1}{1.2}\right)+\left(1-\frac{1}{2.3}\right)+...+\left(1-\frac{1}{2015.2016}\right)\)

=\(1-\frac{1}{1.2}+1-\frac{1}{2.3}+...+1-\frac{1}{2015.2016}\)

=\(\left(1+1+...+1\right)+\left(-\frac{1}{1.2}-\frac{1}{2.3}-...-\frac{1}{2015.2016}\right)\)

=\(2015+\left(-\frac{1}{1}+\frac{1}{2}-\frac{1}{2}+\frac{1}{3}-...-\frac{1}{2015}+\frac{1}{2016}\right)\)

=\(2015+\left(-\frac{1}{1}+\frac{1}{2016}\right)=2015+\left(\frac{-2016}{2016}+\frac{1}{2016}\right)\)

=\(2015+\frac{-2015}{2016}=\frac{4062240}{2016}+\frac{-2015}{2016}=\frac{4060225}{2016}\)