Question

\(M=\left(1-\frac{1}{1.2}\right)+\left(1-\frac{1}{2.3}\right)+...+\left(1-\frac{1}{1995.1996}\right)\)

Answer 1

\(\left(1-\frac{1}{1\cdot2}\right)+\left(1-\frac{1}{2\cdot3}\right)+...+\left(1-\frac{1}{1995\cdot1996}\right)\)

\(=\left(1+1+1+...+1\right)-\left(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{1995\cdot1996}\right)\)

\(=\left(1995\cdot1\right)-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{1995}-\frac{1}{1996}\right)\)

\(=1995-\left(1-\frac{1}{1996}\right)\)

\(=1995-\frac{1995}{1996}\)