Question

Tính \(S=1+\frac{1}{2}\cdot\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+...+\frac{1}{100}\left(1+2+3+...+100\right)\)

Answer 1

\(S=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+...+\frac{1}{100}\left(1+2+3+...+100\right)\)

Ta có công thứ \(1+2+3+...+n=\frac{n\left(n+1\right)}{2}\)

Áp dụng vào bài toán ta được :

\(=1+\frac{1}{2}\cdot\frac{2.3}{2}+\frac{1}{3}\cdot\frac{3.4}{2}+....+\frac{1}{100}\cdot\frac{100.101}{2}\)

\(=1+\frac{3}{2}+\frac{4}{2}+...+\frac{101}{2}\)

\(=\frac{2+3+4+...+101}{2}=\frac{\frac{101.102}{2}-1}{2}=2575\)