Question

Calculati: 1 + 1/1+2 + 1/1+2+3 + ... + 1/1+2+3+...2016 =
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Answer 1

[tex]1+\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+3+4+....+2016}=\\ 1+\frac{1}{\frac{2\cdot 3}{2}}+\frac{1}{\frac{3\cdot 4}{2}}+....+\frac{1}{\frac{2016\cdot 2017}{2}}=\\ 1+2(\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+...+\frac{1}{2016\cdot 2017})=\\ 1+2(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{2016}-\frac{1}{2017})=\\ 1+2(1-\frac{1}{2017})=\\ 1+2\cdot \frac{2016}{2017}=\\ 1+\frac{4032}{2017}=\boxed{\boxed{\frac{6049}{2017}}}[/tex]