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

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

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