S=2/6+2/12+2/20+...+2/90
S=2/2.3+2/3.4+...+2/9.10
S=2(1/2.3+...+1/9.10)
S=2(1/2-1/3+...+1/9-1/10)
S=2.2/5
S=4/5
\(\frac{2}{6}+\frac{2}{12}+\frac{2}{20}+...+\frac{2}{90}\)
\(=\frac{2}{2x3}+\frac{2}{3x4}+\frac{2}{4x5}+...+\frac{2}{9x10}\)
\(=\left(\frac{2}{2x3}+\frac{2}{3x4}+\frac{2}{4x5}+...+\frac{2}{9x10}\right):2x2\)
\(=\left(\frac{1}{2x3}+\frac{1}{3x4}+\frac{1}{4x5}+...+\frac{1}{9x10}\right)x2\)
\(=\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{9}-\frac{1}{10}\right)x2\)
\(=\left(\frac{1}{2}-\frac{1}{10}\right)x2\)
\(=\frac{4}{5}x2\)
\(=\frac{8}{5}\)
