If 3^2 sin 2 α-1, 14 and 3^4 -2 sin 2 α are the first three terms of an A.P. for some α

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 $d=13$

$\mathrm{T}_{6}=\mathrm{a}+5 \mathrm{~d}$

$\mathrm{T}_{6}=1+5 \times 13$

$\mathrm{T}_{6}=66$