Question

 Chứng minh:

         M=3/1^2.2^2+5/2^2.3^2+7/3^2.4^2+......+19/9^2.10^2

Answer 1

         M=3/1^2.2^2+5/2^2.3^2+7/3^2.4^2+......+19/9^2.10^2

\(M=\frac{2^2-1^2}{1^2.2^2}+\frac{3^2-2^2}{2^3.3^2}+\frac{4^2-3^2}{3^2.4^2}+...+\frac{10^2-9^2}{9^2.10^2}\)

\(M=\frac{2^2}{1^2.2^2}-\frac{1^2}{1^2.2^2}+\frac{3^2}{2^2.3^2}-\frac{2^2}{2^2.3^2}+\frac{4^2}{3^2.4^2}-\frac{3^2}{3^2.4^2}+...+\frac{10^2}{9^2.10^2}-\frac{9^2}{9^2.10^2}\)

\(M=1-\frac{1}{2^2}+\frac{1}{2^2}-\frac{1}{3^2}+\frac{1}{3^2}-\frac{1}{4^2}+...+\frac{1}{9^2}-\frac{1}{10^2}\)

\(M=1-\frac{1}{10^2}\)

\(M=\frac{99}{100}\)