Question

Kết quả của tổng M biết: M= \(\dfrac{2}{1x2}\)+\(\dfrac{2}{2x3}\)+......+\(\dfrac{2}{50x51}\)

Answer 1

\(M=\dfrac{1}{2}\left(\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+...+\dfrac{1}{50\times51}\right)\\ M=\dfrac{1}{2}\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{50}-\dfrac{1}{51}\right)\\ M=\dfrac{1}{2}\times\left(1-\dfrac{1}{51}\right)=\dfrac{1}{2}\times\dfrac{50}{51}=\dfrac{25}{51}\)

Answer 2

\(M=\dfrac{2}{1\times2}+\dfrac{2}{2\times3}+...+\dfrac{2}{50\times51}\)

\(M=2\times\left(\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+...+\dfrac{1}{50\times51}\right)\)

\(M=2\times\left(\dfrac{2-1}{1\times2}+\dfrac{3-2}{2\times3}+...+\dfrac{51-50}{50\times51}\right)\)

\(M=2\times\left(\dfrac{2}{1\times2}-\dfrac{1}{1\times2}+\dfrac{3}{2\times3}-\dfrac{2}{2\times3}+...+\dfrac{51}{50\times51}-\dfrac{50}{50\times51}\right)\)

\(M=2\times\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{50}-\dfrac{1}{51}\right)\)

\(M=2\times\dfrac{50}{51}\)

\(M=\dfrac{100}{51}\)