Question

[Nguyễn Hương Giang]

Ko dùng máy tính, hãy tính :

1/1+2 +1/1+2+3 +1/1+2+3+4 +...+1/1+2+3+...+20

Answer 1

[Đoàn Đức Hà]

\(\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+...+20}\)

\(=\frac{2}{2\times3}+\frac{2}{3\times4}+...+\frac{2}{20\times21}\)

\(=2\times\left(\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{20\times21}\right)\)

\(=2\times\left(\frac{3-2}{2\times3}+\frac{4-3}{3\times4}+...+\frac{21-20}{20\times21}\right)\)

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

\(=2\times\left(\frac{1}{2}-\frac{1}{21}\right)\)

\(=\frac{19}{21}\)