Question

[Phạm Thu Vân]

Cho A=1/2x3+1/4x5+1/5x6+1/6x7+1/7x8+1/8x9+.....+1/98x99

Chứng minh rằng 1/5<A<2/5

Answer 1

[Đoàn Đức Hà]

\(A=\frac{1}{2\times3}+\frac{1}{4\times5}+\frac{1}{5\times6}+...+\frac{1}{98\times99}\)

\(=\frac{1}{6}+\frac{5-4}{4\times5}+\frac{6-5}{5\times6}+...+\frac{1}{98\times99}\)

\(=\frac{1}{6}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{98}-\frac{1}{99}\)

\(=\frac{1}{6}+\frac{1}{4}-\frac{1}{99}=\frac{161}{396}>\frac{160}{400}=\frac{2}{5}\)