Question

Chứng minh rằng:

\(y=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}......\frac{99}{100}<\frac{1}{10}\)

Answer 1

gọi 2/3.4/5.6/7.....100/101=B

ta thấy:\(\frac{2}{3}>\frac{1}{2}\)

           \(\frac{4}{5}>\frac{3}{4}\)   

            \(\frac{6}{7}>\frac{5}{6}\)

             .....

          \(\frac{100}{101}>\frac{99}{100}\)

\(\Rightarrow\)\(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{99}{100}>\frac{2}{3}.\frac{4}{5}.\frac{6}{7}.....\frac{100}{101}\)mà:

Y.B=1/2.3/4.5/6.....99/100.2/3.4/5.6/7.....100/101

Y.B=\(\frac{1.3.5.....99.2.4.6....100}{2.4.6....100.3.5.7....101}\)

Y.B=\(\frac{1}{101}\)

vì Y<B \(\Rightarrow\)Y.B>Y.Y

=>Y.Y<\(\frac{1}{101}<\frac{1}{100}\)

=>Y.Y<\(\frac{1}{10}.\frac{1}{10}\)

=>y<1/10(đpcm)