Question

[Nguyễn Minh Quang]

Cho C = 2 + 3/4 + 8/9 + 15/16 + ... + 2499/2500

CMR C > 50

Answer 1

[Xyz OLM]

\(C=1+1+\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{9}\right)+\left(1-\frac{1}{16}\right)+...+\left(1-\frac{1}{2500}\right)\)

\(=\left(1+1+...+1\right)-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{2500}\right)\)

          51 số hạng                                    49 số hạng

= \(51-\left(\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{50.50}\right)\)

\(>51-\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{50.51}\right)=51-\left(\frac{1}{2}-\frac{1}{51}\right)=51-\frac{1}{2}+\frac{1}{51}\)

\(=50,5+\frac{1}{51}>50\left(đpcm\right)\)

Vậy C > 50