Question

a) Cho A=\(\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)......\left(\frac{1}{200}-1\right)\). Hãy so sánh A với \(-\frac{1}{199}\) b) Tính B= \(1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+.....+\frac{1}{500}\left(1+2+3+...+500\right)\)

Answer 1

a) Ta có: \(A=\left(\frac{1}{2}-1\right)\left(\frac{1}{3}-1\right)\left(\frac{1}{4}-1\right)\cdot...\cdot\left(\frac{1}{200}-1\right)\)

\(=\frac{-1}{2}\cdot\frac{-2}{3}\cdot\frac{-3}{4}\cdot...\cdot\frac{-199}{200}\)

\(=-\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot...\cdot\frac{199}{200}\)

\(=-\frac{1}{200}>-\frac{1}{199}\)

Vậy: \(A>-\frac{1}{199}\)

b) Ta có: \(B=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+...+\frac{1}{500}\left(1+2+3+...+500\right)\)

\(=1+\frac{1}{2}\cdot\frac{3\cdot2}{2}+\frac{1}{3}\cdot\frac{4\cdot3}{2}+...+\frac{1}{500}\cdot\frac{501\cdot500}{2}\)

\(=\frac{1}{2}\left(2+3+4+...+501\right)\)

\(=\frac{1}{2}\cdot251000=125500\)

Vậy: B=125500