If the ratio of sum of n terms in two A.P is 2n : n + 1, then the ratio of 8th terms is ……………. 

Correct option is (B) 15 : 8

\(\frac{S_{1n}}{S_{2n}}=\frac{2n}{n+1}\)   ______________(1)   \((\because\) Ratio of sum of n terms of 2 A.P. is given)

Let \(a_1,d_1\) be first term and common difference of first A.P. and \(a_2\;\&\;d_2\) be first term and common difference of second A.P.

\(\therefore\) From (1), we have

\(\frac{\frac n2[2a_1+(n-1)d_1]}{\frac n2[2a_2+(n-1)d_2]}=\frac{2n}{n+1}\)

\(\frac{2a_1+(n-1)d_1}{2a_2+(n-1)d_2}=\frac{2n}{n+1}\)

\(\Rightarrow\frac{a_1+(\frac{n-1}2)d_1}{a_2+(\frac{n-1}2)d_2}=\frac{2n}{n+1}\)    ______________(2)

Let \(\frac{n-1}2=7\) for ratio of \(8^{th}\) terms of both A.P.s.

\(\Rightarrow n-1=14\)

\(\Rightarrow n=14+1=15\)

By putting n = 15 in equation (2), we obtain

\(\frac{a_1+7d_1}{a_2+7d_2}=\frac{2\times15}{15+1}\)

\(=\frac{30}{16}=\frac{15}{8}\)

Hence, the ratio of \(8^{th}\) terms of both A.P. is 15:8.

Correct option is B) 15 : 8