\(\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+...+\dfrac{1}{99\times100}\\ =\dfrac{1}{1}\times\dfrac{1}{2}+\dfrac{1}{2}\times\dfrac{1}{3}+\dfrac{1}{3}\times\dfrac{1}{4}+...+\dfrac{1}{99}\times\dfrac{1}{100}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{99}-\dfrac{1}{100}\\ =1-\dfrac{1}{100}\\ =\dfrac{99}{100}\)
\(\dfrac{1}{1\times2}+\dfrac{1}{2\times3}+...+\dfrac{1}{99\times100}\)
\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{99}-\dfrac{1}{100}\)
\(=1-\dfrac{1}{100}=\dfrac{99}{100}\)
= 2-1/1x2 + 3-2/2x3 + 4-3/3x4 + 5-4/4x5 + .......+100-99/99x100
= 2/1x2 - 1/1x2 + 3/2x3 - 2/2x3 + 4/3x4 - 3/3x4 + 5/4x5 - 4/4x5 +.......+100/99x100 - 99/99x100
=1/1 - 1/2 + 1/2 - 1/3 + 1/3 -1/4 + 1/4 -1/5 +.......+1/100 -1/100
=1/1 - 1/100
=100/100 -1/100
=99/100