Question

Salut! ajutati-ma va rog cu ex.: (1-1/2) + (1-5/6) + (1-11/12) +...+(1-109/110)

Mersi mult

Answer 1

$(1- \frac{1}{2})+(1- \frac{5}{6})+(1- \frac{11}{12})+...+(1- \frac{109}{110})= \\ \\ = \frac{1}{2}+ \frac{1}{6}+ \frac{1}{12}+...+ \frac{1}{110} = \frac{1}{2\cdot1}+ \frac{1}{3\cdot2}+ \frac{1}{4\cdot3}+...+ \frac{1}{11\cdot10} = \\ \\ =\frac{2-1}{2\cdot1}+ \frac{3-2}{3\cdot2}+ \frac{4-3}{4\cdot3}+...+ \frac{11-10}{11\cdot10}= \\ \\ = \frac{2}{2\cdot1}- \frac{1}{2\cdot1} +\frac{3}{3\cdot2}- \frac{2}{3\cdot2} +\frac{4}{4\cdot3}- \frac{3}{4\cdot3} +...+\frac{11}{11\cdot10}- \frac{10}{11\cdot10}=$

$=1- \frac{1}{2}+ \frac{1}{2}-\frac{1}{3}+ \frac{1}{3}-\frac{1}{4}+\frac{1}{4}-...-\frac{1}{9}+ \frac{1}{9}-\frac{1}{10}+ \frac{1}{10}- \frac{1}{11} = \\ \\ = 1- \frac{1}{11}= \frac{10}{11}$