### If 18th and 11th term of an A.P. are in the ratio 3 : 2, then its 21st and 5th terms are in the ratio?

A. 3 : 2 B. 3 : 1 C. 1 : 3 D. 2 : 3 Answer: Option B
$\begin{array}{l}18^{\text{th }}\text{ term }:11^{\text{th }}\text{ term }=3:2\\ \Rightarrow\Large\frac{a_{18}}{a_{11}}=\frac{3}{2}\\ \Rightarrow\Large\frac{a+17d}{a+10d}=\frac{3}{2}\\ \Rightarrow2a+34d=3a+30d\\ \Rightarrow34d-30d=3a-2a\\ \Rightarrow a=4d\end{array}$ Now, \begin{aligned}\frac{a_{21}}{a_5}&=\frac{a+20d}{a+4d}\\ &=\frac{4d+20d}{4d+4d}\\ &=\frac{24d}{8d}\\ &=\frac{3}{1}\\ \therefore a_{21}&:a_5=3:1\end{aligned}