Question

1) Cho 

1/M=1/1+2+3+1/1+2+3+4+..+1/1+2+3+..+59

CMR M>2/3

Answer 1

phải là M<2/3 mới giải đc

\(M=\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+59}\)

\(=\frac{1}{\left(3+1\right).3:2}+\frac{1}{\left(4+1\right).4:2}+...+\frac{1}{\left(59+1\right).59:2}\)

\(=\frac{1}{6}+\frac{1}{10}+...+\frac{1}{1770}\)

\(=\frac{2}{12}+\frac{2}{20}+...+\frac{2}{3540}\)

\(=2\left(\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{595.60}\right)\)

\(=2\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{59}-\frac{1}{60}\right)\)

\(=2\left(\frac{1}{3}-\frac{1}{60}\right)\)

\(=\frac{2}{3}-\frac{2}{60}< \frac{2}{3}\)