Question

Find the value of $\frac{50}{72}+\frac{50}{90}+\frac{50}{110}+\frac{50}{132}+...+\frac{50}{9900}$.

Answer 1

$\frac{50}{72}+\frac{50}{90}+....+\frac{50}{9900}$

$=50(\frac{1}{72}+\frac{1}{90}+..+\frac{1}{9900})$

$=50(\frac{1}{8\times 9}+\frac{1}{10\times 9}+...+\frac{1}{99\times 100})$

$=50(\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10}+\frac{1}{10}-\frac{1}{11}+...+\frac{1}{99}-\frac{1}{100})$

$=50(\frac{1}{8}-\frac{1}{100})=50\left(\frac{100-8}{800}\right)$

$=\frac{50\times 92}{800}=\frac{23}{4}$