If the arithmetic progression whose common difference is nonzero the sum of first `3n` terms is equal to the sum of next `n` terms. Then, find the ratio of the sum of the `2n` terms to the sum of next `2n` terms.

Options

A. `(1)/(5)`
B. `(2)/(3)`
C. `(3)/(4)`
D. None of these

Explanation

Given, common difference`ne0`
`S_(3n)=S_(4n)-S_(3n)`
`implies2*S_(3n)=S_(4n)" " [" let " S_(n)=Pn^(2)+Qn]`
`implies2*[P(3n)^(2)+Q(3n)]=P(4n)^(2)+Q(4n)`
`implies 2Pn^(2)+2Qn=0`
or `Q=-nP" " "….(i)"`
`:.(S_(2n))/(S_(4n)-S_(2n))= (P(2n)^(2)+Q(2n))/([P(4n)^(2)+Q(4n)]-[P(2n)^(2)+Q(2n)])`
`=(2n(2nP+Q))/(12Pn^(2)+2nQ)=(2nP+Q)/(6nP+Q)`
`=(2nP-nP)/(6nP-nP)=(1)/(5)" " [" from Eq. (i) "]`.