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 ,…

If $a, b, c$ are in AP, then $b$ is A.M of $a \& c$. 

$\therefore 2 b=a+c$

$\Rightarrow 28=3^{2\sin 2\alpha -1}+3^{4-2\sin 2\alpha }$

Putting, $3^{2\sin 2\alpha }=x$ we get,

$28=\frac{x}{3}+\frac{81}{x}\Rightarrow x^2-84x+243=0$

$\Rightarrow (x-3)(x-81)=0$

$\therefore 3^{2\sin 2\alpha }=3$ or $3^4$

$\sin 2\alpha =\frac{1}{2}$, $\because \sin 2\alpha \ne 2$

Terms are $1,14,27,\dots \dots$ then

$T_6=1+5\left(13\right)=66$