Question

Chứng minh rằng 1/20.23+1/23.26+1/2.29+...+1/77.80<1/9

Ai giúp mình tìm thêm một số cách giải, nhé?

Answer 1

Ta có: \(\frac{1}{20.23}+\frac{1}{23.26}+\frac{1}{26.29}+...+\frac{1}{77.80}\)

\(=\frac{1}{3}.\left[\left(\frac{1}{20}-\frac{1}{23}\right)+\left(\frac{1}{23}-\frac{1}{26}\right)+\left(\frac{1}{26}-\frac{1}{29}\right)+...+\left(\frac{1}{77}-\frac{1}{80}\right)\right]\)

\(=\frac{1}{3}.\left(\frac{1}{20}-\frac{1}{80}\right)\)

\(=\frac{1}{3}.\frac{3}{80}=\frac{1}{80}\) \(< \frac{1}{9}\)

Vậy \(\frac{1}{20.23}+\frac{1}{23.26}+\frac{1}{26.29}+...+\frac{1}{77.80}< \frac{1}{9}\)