Question

Tính nhanh :

\(A=\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\dfrac{1}{1+2+3+4}+...+\dfrac{1}{1+2+3+...+10}\)

Answer 1

\(\dfrac{1}{1+2}+\dfrac{1}{1+2+3}+\dfrac{1}{1+2+3+4}+...+\dfrac{1}{1+2+3+...+10}\)

\(=\dfrac{1}{\dfrac{2.3}{2}}+\dfrac{1}{\dfrac{3.4}{2}}+\dfrac{1}{\dfrac{4.5}{2}}+...+\dfrac{1}{\dfrac{10.11}{2}}\)

\(=\dfrac{2}{2.3}+\dfrac{2}{3.4}+\dfrac{2}{4.5}+...+\dfrac{2}{10.11}\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{10}-\dfrac{1}{11}\right)\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{11}\right)\)

\(=2\cdot\dfrac{9}{22}\)

\(=\dfrac{9}{11}\)