Find the sum of n terms of the series
`(1)/((2 xx 5))+(1)/((5xx8))+(1)/((8xx11))+... .`

Explanation

We have
`T_(k)=(1)/(("kth term of "2,5,8, ...) xx("kth term of "5,8,11, ...))`
`=(1)/({2+(k-1)xx3}xx{5+(k-1)xx3})`
`=(1)/((3k-1)(3k+2))=(1)/(3){(1)/((3k-1))-(1)/((3k+2))}`.
`therefore T_(k)=(1)/(3){(1)/((3k-1))-(1)/((3k+2))}. " " `...(i)
Putting `k=1,2,3, ..., n` successively in (i), we get
`T_(1)=(1)/(3)((1)/(2)-(1)/(5))`
`T_(2)=(1)/(3)((1)/(5)-(1)/(8))`
`T_(3)=(1)/(3)((1)/(8)-(1)/(11))`
... ... ... ...
... ... ... ...
`T_(n) =(1)/(3){(1)/((3n-1))-(1)/((3n+2))}.`
Adding columnwise, we get
`S_(n)=(T_(1)+T_(2)+T_(3)+...+T_(n))`
`=(1)/(3)((1)/(2)-(1)/(3n+2))=(n)/(2(3n+2)).`