Question

Xét tổng S gồm 20 số hạng:

S=\(\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+.......+\frac{1}{20.21.22.23}\)

Answer 1

\(S=\frac{1}{1.2.3.4}+\frac{1}{2.3.4.5}+.....+\frac{1}{20.21.22.23}\)

\(=\frac{1}{3}\left(\frac{3}{1.2.3.4}+\frac{3}{2.3.4.5}+......+\frac{3}{20.21.22.23}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2.3}-\frac{1}{2.3.4}+\frac{1}{2.3.4}-\frac{1}{3.4.5}+.....+\frac{1}{20.21.22}-\frac{1}{21.22.23}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2.3}-\frac{1}{21.22.23}\right)\)

\(=\frac{1}{2}.\frac{295}{1771}=\frac{295}{3542}\)