Question

Cho A = 1/2.3/4.5/6...99/100.

Chứng minh :

a) A <1/10

b) A >1/15

Answer 1

\(A=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}\)

\(\Rightarrow A< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{100}{101}\)

\(\Rightarrow A^2< \frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{100}{101}.\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}\)

\(\Rightarrow A^2< \frac{1}{101}< \frac{1}{100}=\frac{1}{10^2}\)

\(\Leftrightarrow A< \frac{1}{10}\)

Answer 2

\(A=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}\)

\(\Rightarrow A>\frac{1}{2}.\frac{2}{3}.\frac{4}{5}...\frac{98}{99}\)

\(\Rightarrow A^2>\frac{1}{2}.\frac{2}{3}.\frac{4}{5}...\frac{98}{99}.\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{99}{100}\)

\(\Rightarrow A^2>\frac{1}{200}>\frac{1}{225}=\frac{1}{15^2}\)

\(\Rightarrow A>\frac{1}{15}\)

Answer 3

A=1/2 . 3/4 . 5/6 ... 99/100 < 2/3.4/5.6/7....100/101
Đặt 2/3.4/5.6/7....100/101=C
=>A<C
=>A.A<A.C=1/2 . 3/4 . 5/6 ... 99/100. 2/3.4/5.6/7....100/101=1/101
=>A.A<1/101<1/100
=>A<1/10
hay A<B
vậy A<B