Question

Tính tổng: 1/1+2+3 +1/1+2+3+4 +... 1/1+2+3+4+5+...+59

Answer 1

\(\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+\cdots+\frac{1}{1+2+3+\cdots+59}\)

\(=\frac{1}{3\cdot\frac42}+\frac{1}{4\cdot\frac52}+\cdots+\frac{1}{59\cdot\frac{60}{2}}\)

\(=\frac{2}{3\cdot4}+\frac{2}{4\cdot5}+\cdots+\frac{2}{59\cdot60}\)

\(=2\left(\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\cdots+\frac{1}{59\cdot60}\right)\)

\(=2\left(\frac13-\frac14+\frac14-\frac15+\cdots+\frac{1}{59}-\frac{1}{60}\right)\)

\(=2\left(\frac13-\frac{1}{60}\right)=2\cdot\frac{19}{60}=\frac{19}{30}\)