Question

Найдите значения выражения:
а) 1/6+1/12+1/20+...+1/380
б) 2/15+2/35+2/63+...+2/323

Answer 1

$frac{1}{6} + frac{1}{12} + frac{1}{20} +...+ frac{1}{380} = frac{1}{2*3} + frac{1}{3*4} + frac{1}{4*5} +...+ frac{1}{19*20}=\= frac{1}{2} - frac{1}{3} + frac{1}{3} -frac{1}{4} + frac{1}{4} - frac{1}{5} +...+ frac{1}{18} - frac{1}{19} + frac{1}{19} - frac{1}{20} = frac{1}{2} - frac{1}{20} = frac{9}{20}=\=0,45$

$frac{2}{15} + frac{2}{35} + frac{2}{63} +...+ frac{2}{323} = frac{2}{3*5} + frac{2}{5*7} + frac{2}{7*9} +...+ frac{2}{17*19} =\= frac{1}{3} - frac{1}{5} + frac{1}{5} - frac{1}{7} + frac{1}{7} - frac{1}{9} +...+ frac{1}{15} - frac{1}{17} + frac{1}{17} - frac{1}{19} = frac{1}{3} - frac{1}{19} = frac{16}{57}$