Question

 Tính

\(\frac{2.2016}{1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+..+2016}}\)

Answer 1

Mẫu số = \(1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+...+2016}\)

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

\(=1+\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{1}{2016.2017}\)

\(=2.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2016.2017}\right)\)

\(=2.\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2016}-\frac{1}{2017}\right)\)

\(=2.\left(1-\frac{1}{2017}\right)\)

\(=\frac{2.2016}{2017}\)

Vậy phân số đề bài cho \(=\frac{2.2016}{\frac{2.2016}{2017}}=2.2016.\frac{2017}{2.2016}=2017\)