a) $M = \frac{2}{3\times 5}+\frac{2}{5\times 7}+\frac{2}{7\times 9}+...+\frac{2}{97\times 99}$.

$M = \frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{97}-\frac{1}{99}$.

$M = \frac{1}{3}-\frac{1}{99}$.

$M = \frac{33}{99}-\frac{1}{99}$.

$M = \frac{32}{99}$.

b) $N = \frac{3}{5\times 7}+\frac{3}{7\times 9}+\frac{3}{9\times 11}+...+\frac{3}{197\times 199}$

$N = \frac{3}{2}(\frac{2}{5\times 7}+\frac{2}{7\times 9}+\frac{2}{9\times 11}+...+\frac{2}{197\times 199})$

$N = \frac{3}{2}(\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{197}-\frac{1}{199})$

$N = \frac{3}{2}(\frac{1}{5}-\frac{1}{199})$

$N = \frac{3}{2}(\frac{199}{995}-\frac{5}{995})$

$N = \frac{3}{2}\frac{194}{995}$

$N = \frac{291}{995}$

c) $P =  \frac{1}{1\times 2}+\frac{2}{2\times 4}+\frac{3}{4\times 7}+...+\frac{10}{46\times 56}$

$P =  \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+...+\frac{1}{46}-\frac{1}{56}$

$P =  1 - \frac{1}{56}$

$P =  \frac{55}{56}$