Question

[Công chúa phép thuật Bloom]

C = 1/2 x 3/4 x 5/6 x ... x 199/200

Chứng tỏ C < 1/14

Answer 1

[Đoàn Đức Hà]

\(C=\frac{1}{2}\times\frac{3}{4}\times\frac{5}{6}\times...\times\frac{199}{200}\)

\(C^2=\left(\frac{1}{2}\right)^2\times\left(\frac{3}{4}\right)^2\times\left(\frac{5}{6}\right)^2\times...\times\left(\frac{199}{200}\right)^2\)

\(< \frac{1}{2}\times\frac{2}{3}\times\frac{3}{4}\times\frac{4}{5}\times\frac{5}{6}\times\frac{6}{7}\times...\times\frac{199}{200}\times\frac{200}{201}\)

\(=\frac{1}{201}< \frac{1}{196}\)

\(\Rightarrow C< \sqrt{\frac{1}{196}}=\frac{1}{14}\)