Question

Chứng minh rằng :

\(\frac{1}{1x2}\) + \(\frac{1}{3x4}\) + .... + \(\frac{1}{49x50}\)= \(\frac{1}{26}\)+ \(\frac{1}{27}\)+....+ \(\frac{1}{50}\)

Answer 1

   \(\frac{1}{1x2}+\frac{1}{3x4}+....+\frac{1}{49x50}\)

\(=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+......+\frac{1}{49}-\frac{1}{50}\)

\(=\left(1+\frac{1}{3}+...+\frac{1}{49}\right)+\left(-\frac{1}{2}-\frac{1}{4}-.....-\frac{1}{50}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+......+\frac{1}{49}+\frac{1}{50}\right)-2\left(\frac{1}{2}+\frac{1}{4}......+\frac{1}{50}\right)\)

\(=\left(1+\frac{1}{2}+\frac{1}{3}+.....+\frac{1}{49}+\frac{1}{50}\right)-\left(1+\frac{1}{2}+......+\frac{1}{25}\right)\)

\(=\frac{1}{26}+\frac{1}{27}+....+\frac{1}{50}\left(đpcm\right)\)