\(\frac{10}{2\cdot3}+\frac{10}{3\cdot4}+\frac{10}{4\cdot5}+...+\frac{10}{8\cdot9}\)
\(=10\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{8}-\frac{1}{9}\right)\)
\(=10\cdot\left(\frac{1}{2}-\frac{1}{9}\right)=10\cdot\frac{7}{18}=\frac{35}{9}\)
\(=10\cdot\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{8}-\frac{1}{9}\right)\)
\(=10\cdot\left(\frac{1}{2}-\frac{1}{9}\right)\)
\(=10\cdot\frac{7}{9}\)
\(=\frac{70}{9}\)
10/2x3 + 10/3x4 + 10/4x5 + 10/5x6 + 10/6x7 + 10/7x8 + 10/8x9
= 10 x ( 1/2x3 + 1/3x4 + 1/4x5 + 1/5x6 + 1/6x7 + 1/7x8 + 1/8x9 )
= 10 x ( 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + 1/5 - 1/6 + 1/6 - 1/7 + 1/7 - 1/8 + 1/8 - 1/9 )
= 10 x ( 1/2 - 1/9 )
= 10 x 7/18
= 35/9
