Question

Найти рациональный способ решения примера: 1/1*2+1/2*3+1/3*4+.....+1/49*50

Answer 1

$\frac{1}{1*2}+\frac{1}{2*3}+...+\frac{1}{49*50}=$
$\frac{2-1}{1*2}+\frac{3-2}{2*3}+...+\frac{50-49}{49*50}=$
$\frac{2}{1*2}-\frac{1}{1*2}+\frac{3}{2*3}-\frac{2}{2*3}+....+\frac{50}{49*50}-\frac{49}{49*50}$
$1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}=$
$1-\frac{1}{50}=1-0.02=0.98$

$\frac{1}{k(k+1)}=\frac{(k+1)-k}{k(k+1)}=\frac{1}{k}-\frac{1}{k+1}$