JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Sequences and Series

For the three to form an AP, twice the middle term equals the sum of the outer two: $28=3^{2\sin 2\alpha -1}+3^{4-2\sin 2\alpha }$. Let $t=3^{2\sin 2\alpha }$; then $28=\frac{t}{3}+\frac{81}{t}$, that is $t^2-84t+243=0$, which factors as $(t-3)(t-81)=0$. The root $t=81$ would need $2\sin 2\alpha =4$, impossible, so $t=3$ and $2\sin 2\alpha =1$. The three terms are then $1,14,27$, an AP of common difference $13$, and the sixth term is $1+5\times 13=66$.