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  

$3^{2\sin 2\alpha -1},14,3^{4-2\sin 2\alpha }$ are in A.P 

$\Rightarrow 2b=a+c\Rightarrow 2\left(14\right)=3^{2\sin 2\alpha -1}+3^{4-2\sin 2\alpha }\to \left(1\right)$

Put  $t=3^{2\sin 2\alpha }$

$\left(1\right)\Rightarrow 28=t\left[\frac{1}{3}\right]+\frac{81}{t}$

$\Rightarrow t^2-84t+243=0$

$\left(t-3\right)\left(t-81\right)=0\Rightarrow t=3\left(or\right)t=81$

i.e  $\begin{array}{c}{3}^{2\sin 2\alpha }=3^1\\ 2\sin 2\alpha =1\end{array}\left|\begin{array}{l}{3}^{2\sin 2\alpha }=81=3^4\\ 2\sin 2\alpha =4\end{array}\right.$

$1^{st}$ term  $a=3^{2\sin 2\alpha -1}=3^{2\left(\frac{1}{2}\right)-1}=3^o=1$

$2^{nd}$term =14

Common difference d=14-1=13

$$$6^{th}$term = $a+5d=1+5\left(13\right)=66$