Question

\(\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+.....+\frac{1}{1+2+3+4+.......+15}\)

Answer 1

\(\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+4+...+15}\)

\(=\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{220}\)

\(=\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{15.16}\)

\(=2\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{15}-\frac{1}{16}\right)\)

\(=2\cdot\left(\frac{1}{2}-\frac{1}{16}\right)=2\cdot\frac{7}{16}=\frac{7}{8}\)

Answer 2

Mình cảm ơn bạn nhiều nhé