Question

Tính \(A=\frac{11}{1.2.3}+\frac{11}{2.3.4}+\frac{11}{3.4.5}+...+\frac{11}{98.99.100}\)

Answer 1

\(A=\frac{11}{1.2.3}+\frac{11}{2.3.4}+\frac{11}{3.4.5}+...+\frac{11}{98.99.100}\)

\(A=\frac{11}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\right)\)

\(A=\frac{11}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

\(A=\frac{11}{2}.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)=\frac{11}{2}.\left(\frac{1}{2}-\frac{1}{9900}\right)=\frac{11}{2}.\left(\frac{4950}{9900}-\frac{1}{9900}\right)=\frac{11}{2}.\frac{4949}{9900}=\frac{4949}{1800}\)