If the sum of the n terms of a G.P. is `(3^(n)-1)`, then find the sum of the series whose terms are reciprocal of the given G.P..

Explanation

We have sum of n terms, `S_(n)=3^(n)-1` or `S_(r)=3^(r )-1`
`therefore T_(r)=S_(r)-S_(r-1)=(3^(r )-1)-(3^(r-1)-1)`
`=3^(r-1)(3-1)=2xx3^(r-1)`
`therefore1/(T_(r))=1/2cdot(1/3)^(r-1)`
So, series formed by reciprocals of the given G.P is
`1/2,1/2xx(1/3),1/2xx(1/3)^(2),....`
`therefore` Sum of n terms `=(1/2(1-(1/3)^(n)))/(1-1/3)=3/4(1-(1/3)^(n))`