Question:
If $3^{2} \sin 2 \alpha-1,14$ and $3^{4-2} \sin 2 \alpha$ are the first three terms of an A.P. for some $\alpha$, then the sixth term of this A.P. is:
Correct Option: 1
Solution:
Given that
$3^{4-\sin 2 \alpha}+3^{2} \sin 2 \alpha-1=28$
Let $3^{2} \sin 2 a=\mathrm{t}$
$\frac{81}{t}+\frac{t}{3}=28$
$t=81,3$
$3^{2} \sin 2 \alpha=3^{1}, 3^{4}$
$2 \sin 2 \alpha=1,4$
$\sin 2 \alpha=\frac{1}{2}, 2$ (rejected)
First term $a=3^{2} \sin 2 \alpha-1$
$a=1$
Second term $=14$
$\therefore$ common difference $d=13$
$\mathrm{T}_{6}=\mathrm{a}+5 \mathrm{~d}$
$\mathrm{T}_{6}=1+5 \times 13$
$\mathrm{T}_{6}=66$