Question

\(1+\frac{1}{2}.\left(1+2\right)+\frac{1}{3}.\left(1+2+3\right)+...+\frac{1}{20}.\left(1+2+3+...+20\right)\\\)

Tính càng nhanh càng tốt

nhanh và đúng nhất mình k 2 lần~~~~

Answer 1

Từ công thức:\(1+2+........+n=\frac{n.\left(n+1\right)}{2}\)

Cho \(n\in\)N*.CMR:\(\frac{1}{n}.\left(1+2+...+n\right)=\frac{n+1}{2}\)

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

Ta có:\(1+\frac{1}{2}\left(1+2\right)+......+\frac{1}{20}.\left(1+2+.....+20\right)\)

\(=1+\frac{1}{2}.\frac{2\left(2+1\right)}{2}+\frac{1}{3}.\frac{3.\left(3+1\right)}{2}+........+\frac{1}{20}.\frac{20\left(20+1\right)}{2}\)

\(=1+\frac{3}{2}+...............+\frac{21}{2}\)

\(=\frac{2+3+......+21}{2}\)

\(=\frac{230}{2}=165\)