Question

(1/2*3+1/3*4+1/4*5+...+1/63*64)+(-75/128)

Answer 1

$\frac{1}{k \cdot (k+1)} = \frac{(k+1)-k}{k(k+1)}= \frac{k+1}{k(k+1)}- \frac{k}{k(k+1)}= \frac{1}{k}- \frac{1}{k+1}. \\ ------------------------------ \\ \\ S=( \frac{1}{2 \cdot 3}+ \frac{1}{3 \cdot 4}+ \frac{1}{4 \cdot 5}+...+ \frac{1}{63 \cdot 64})-\frac{75}{128} = \\ \\ ~~~= ( \frac{1}{2} - \frac{1}{3}+ \frac{1}{3} - \frac{1}{4}+ \frac{1}{4}- \frac{1}{5}+...+ \frac{1}{63}- \frac{1}{64})- \frac{75}{128}= \\ \\ ~~~= (1- \frac{1}{64})- \frac{75}{128}=$

$~~~= \frac{128}{128}- \frac{2}{128}- \frac{75}{128}= \\ \\ = \frac{51}{128}$

Answer 2

[1/2-1/3 +1/3-1/4 +1/4-1/5 +....+1/63-1/64]+[-75/128]
⇒[1/2-1/64]-75/128
⇒[64/128-2/128]-75/128
⇒62/128-75/128=-13/128⇒fractie ireductibila;