Question

Вычислите: 1/9*11+1/11*13+1/13*15+...+1/23*25

Answer 1

$\frac{1}{9*11}+\frac{1}{11*13}+\frac{1}{13*15}+...+\frac{1}{23*25}=\\\\\frac{1}{2}*\frac{11-9}{9*11}+\frac{1}{2}*\frac{13-11}{11*13}+\frac{1}{2}*\frac{15-13}{13*15}+...+\frac{1}{2}*\frac{25-23}{23*25}=\\\\\frac{1}{2}*(\frac{1}{9}-\frac{1}{11}+\frac{1}{11}-\frac{1}{13}+\frac{1}{13}-\frac{1}{15}+...+\frac{1}{23}-\frac{1}{25})=\\\\\frac{1}{2}*\frac{25-9}{9*25}=\frac{8}{225}$