Question

Решите пожайлуста.....

Answer 1

$1)~~~displaystyle frac{1}{1cdot2}+ frac{1}{2cdot3}+frac{1}{3cdot4}+...+frac{1}{99cdot100}=\\ =frac{2-1}{1cdot2}+ frac{3-2}{2cdot3}+frac{4-3}{3cdot4}+...+frac{100-99}{99cdot100}=\ \ =1- frac{1}{2} + frac{1}{2} - frac{1}{3} + frac{1}{3} - frac{1}{4} +...+ frac{1}{99}- frac{1}{100} =1- frac{1}{100} = frac{99}{100}$


$2)~~~ displaystyle frac{1}{2cdot4} +frac{1}{4cdot6} +frac{1}{6cdot8} +...+frac{1}{98cdot100} =\ \ \ = frac{1}{2} cdot bigg( frac{1}{1cdot2}+ frac{1}{2cdot3}+ frac{1}{3cdot4}+...+ frac{1}{49cdot50}bigg)=\ \ \ = frac{1}{2}cdotbigg(1- frac{1}{2} + frac{1}{2} -frac{1}{3} +frac{1}{3}-frac{1}{4}+...+frac{1}{49}-frac{1}{50} bigg)= frac{1}{2} cdotbigg(1-frac{1}{50}bigg)= frac{49}{100}$