Question

Вычислите сумму 1 / 1*3 +1 /3*5+...........+ 1 / 123*125

Answer 1

$\frac{1}{1*3} + \frac{1}{3*5} +...+ \frac{1}{123*125} = \\ \\ \frac{1}{2} *(1- \frac{1}{3} )+ \frac{1}{2} *( \frac{1}{3} - \frac{1}{5} )+...+ \frac{1}{2} *( \frac{1}{123} - \frac{1}{125} )= \\ \\ \frac{1}{2} *(1- \frac{1}{3} + \frac{1}{3} - \frac{1}{5} +...+ \frac{1}{123} - \frac{1}{125})= \\ \\ \frac{1}{2} *(1- \frac{1}{125} )= \frac{1}{2} * \frac{124}{125} = \frac{62}{125}$