\(\frac{1}{18}+\frac{1}{54}+\frac{1}{108}+...+\frac{1}{990}\)
\(=\frac{1}{3}\cdot\left(\frac{3}{3\cdot6}+\frac{3}{6\cdot9}+\frac{3}{9\cdot12}+...+\frac{3}{30\cdot33}\right)\)
\(=\frac{1}{3}\cdot\left(\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+...+\frac{1}{30}-\frac{1}{33}\right)\)
\(=\frac{1}{3}\cdot\left(\frac{1}{3}-\frac{1}{33}\right)\)
\(=\frac{1}{3}\cdot\frac{10}{33}\)
\(=\frac{10}{99}\)
1/3x(3/3x6+3/6x9+3/9x12+...+3/30x33)
1/3x(1/3x6+1/6x9+1/9x12+...+1/30x33)
1/3x(1/3-1/6)+(1/6-1/9)+(1/9-1/12)+...+(1/10-1/33)
1/3x(1/3-1/33)
1/3x10/33
10/99
