Question

Find the sum of the first 100 terms in the series


`(1)/((1*2)) + (1)/((2*3)) + (1)/((3*4)) + . . . (1)/(n*(n+1))`

Answer 1

Hello,


`(1)/(n) - (1)/(n+1) = (1)/(n(n+1)) \\ (1)/(1*2) = (1)/(1) - (1)/(2) \\ (1)/(2*3) = (1)/(2) - (1)/(3) \\ (1)/(3*4) = (1)/(3) - (1)/(4) \\ ...\\ (1)/(n*(n+1)) = (1)/(n) - (1)/(n+1) \\`

Adding member by member, we have


`(1)/(1*2) + (1)/(2*3) +(1)/(3*4) +...(1)/(n*(n+1))=\\ (1)/(1) - (1)/(n+1) \\ = (n)/(n+1) \\`

if n=100 sum
`\boxed{= (100)/(101) }`