[tex] \frac{1}{k \cdot (k+1)} = \frac{(k+1)-k}{k(k+1)}= \frac{k+1}{k(k+1)}- \frac{k}{k(k+1)}= \frac{1}{k}- \frac{1}{k+1}. \\ ------------------------------ \\ \\ S=( \frac{1}{2 \cdot 3}+ \frac{1}{3 \cdot 4}+ \frac{1}{4 \cdot 5}+...+ \frac{1}{63 \cdot 64})-\frac{75}{128} = \\ \\ ~~~= ( \frac{1}{2} - \frac{1}{3}+ \frac{1}{3} - \frac{1}{4}+ \frac{1}{4}- \frac{1}{5}+...+ \frac{1}{63}- \frac{1}{64})- \frac{75}{128}= \\ \\ ~~~= (1- \frac{1}{64})- \frac{75}{128}= [/tex]
[tex]~~~= \frac{128}{128}- \frac{2}{128}- \frac{75}{128}= \\ \\ = \frac{51}{128} [/tex]
