Question

[ggevkhs]

1/1*2*3+1/2*3*4+...+1/18*19*20<1/4

Answer 1

[Nguyễn Phương Uyên]

đặt \(A=\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+...+\frac{1}{18\cdot19\cdot20}\)

\(A=\frac{1}{2}\left(\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+...+\frac{2}{18\cdot19\cdot20}\right)\)

\(A=\frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{18\cdot19}-\frac{1}{19\cdot20}\right)\)

\(A=\frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{19\cdot20}\right)\)

\(A=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{380}\right)\)

\(A=\frac{1}{2}\cdot\frac{189}{380}\)

\(A=\frac{189}{760}< \frac{1}{4}\)