Question

\(\frac{2017}{1\times2\text{×}3}+\frac{2017}{2\text{×}3\text{×}4}+\frac{2017}{3\text{×}4\text{×}5}+..+\frac{2017}{19\text{×}20\text{×}21}\)

Answer 1

\(\frac{2017}{1.2.3}+\frac{2017}{2.3.4}+\frac{2017}{3.4.5}+...+\frac{2017}{19.20.21}\)

\(=2017\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{19.20.21}\right)\)

Answer 2

\(=2017.\left(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{19.20.21}\right)\)

\(=2017.\left(1-\frac{1}{2}-\frac{1}{3}-\left(\frac{1}{2}-\frac{1}{3}-\frac{1}{4}\right)-...-\left(\frac{1}{19}-\frac{1}{20}-\frac{1}{21}\right)\right)\)

\(=2017.\left(1+\frac{1}{21}\right)\)phá ngoặc trước dấu trừ đổi dấu,rút gọn:

\(=2017.\frac{20}{21}=\frac{40340}{21}\)