\(2C=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+\dfrac{2}{98.99.100}\)
\(=\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+...+\dfrac{1}{98.99}-\dfrac{1}{99.100}\)
\(=\dfrac{1}{1.2}-\dfrac{1}{99.100}=\dfrac{50.99-1}{100.99}=\dfrac{4949}{9900}\)
`A=1/[1.2.3]+1/[2.3.4]+....+1/[98.99.100]`
`A=1/2.(2/[1.2.3]+2/[2.3.4]+....+2/[98.99.100])`
`A=1/2.(1/[1.2]-1/[2.3]+1/[2.3]-1/[3.4]+....+1/[98.99]-1/[99.100])`
`A=1/2.(1/[1.2]-1/[99.100])`
`A=1/2.(1/2-1/9900)`
`A=1/2.(4950/9900-1/9900)`
`A=1/2 . 4949/9900`
`A=4949/19800`
\(C=\frac{1}{1.2.3}+\frac{1}{2.3.4}+\frac{1}{3.4.5}+...+\frac{1}{98.99.100}\)
\(C=\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+\frac{2}{3.4.5}+...+\frac{2}{98.99.100}\right)\)
\(C=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+\frac{1}{3.4}-\frac{1}{4.5}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)
\(C=\dfrac{1}{2}.\left(\dfrac{1}{2}-\dfrac{1}{9900}\right)\)
\(C=\dfrac{1}{2}.\dfrac{4949}{9900}=\dfrac{4949}{19800}\)