Question

\(\dfrac{1}{1. 2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{999.1000}\)

Answer 1

\(\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{999\cdot1000}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{999}-\dfrac{1}{1000}\\ =1-\dfrac{1}{1000}=\dfrac{999}{1000}\)

Answer 2

`1/(1.2) + 1/(2.3) + ... + 1/(999.1000)`

`= 1 - 1/2 + 1/2 - 1/3 + ... + 1/999 - 1/1000`

`= 1- 1/1000`

`= 1000/1000 - 1/1000`

`= 999/1000