\(\frac{8}{35}+\frac{8}{63}+\frac{8}{99}+...+\frac{8}{575}\)
= \(4x\left(\frac{2}{3x5}+\frac{2}{5x7}+\frac{2}{7x9}+...+\frac{2}{23x25}\right)\)
= \(4x\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{23}-\frac{1}{25}\right)\)
=\(4x\left(\frac{1}{3}-\frac{1}{25}\right)\)
= \(4x\frac{22}{75}\)
=\(\frac{88}{75}\)
=\(4.\left(\frac{2}{5.7}+\frac{2}{7.9}+\frac{2}{9.11}+...+\frac{2}{23.25}\right)\)
=\(4.\left(\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{23}-\frac{1}{25}\right)\)
=\(4.\left(\frac{1}{5}-\frac{1}{25}\right)\)
=\(4.\frac{22}{75}\)
=\(\frac{88}{75}\)
