Question

Найти значение выражения
$\displaystyle \frac{1}{2}+\frac{1}{6} +\frac{1}{12} +\frac{1}{20} +\ldots+\frac{1}{132} =?$

Answer 1

Ответ:

$\dfrac{11}{12}$

Объяснение:

$\displaystyle\bf\\\frac{1}{2} =1-\frac{1}{2}\\\\\\\frac{1}{6}=\frac{3-2}{6} =\frac{3}{6} -\frac{2}{6} =\frac{1}{2} -\frac{1}{3} \\\\\\\frac{1}{12}=\frac{4-3}{12} =\frac{4}{12} -\frac{3}{12} =\frac{1}{3} -\frac{1}{4} \\\\\\\frac{1}{20}=\frac{5-4}{6} =\frac{5}{20} -\frac{4}{20} =\frac{1}{4} -\frac{1}{5} \\\\...\\\\\frac{1}{132}=\frac{12-11}{132} =\frac{12}{132} -\frac{11}{132} =\frac{1}{11} -\frac{1}{12} \\\\\\\frac{1}{2} +\frac{1}{6} +\frac{1}{12}+\frac{1}{20} +...+\frac{1}{132}=$

$\displaystyle\bf\\1-\frac{1}{2} +\frac{1}{2} -\frac{1}{3}+\frac{1}{3} -\frac{1}{4}+\frac{1}{4} -\frac{1}{5}+....+\frac{1}{11} -\frac{1}{12}=\\\\\\=1-\frac{1}{12} =\frac{11}{12}$