$\text{ The numbers }{3}^{2\sin 2\alpha -1},14\text{ and }{3}^{4-2\sin 2\alpha }\text{ from first three terms of an A.P its }{5}^{\text{th }}\text{ term is}$

$$\begin{gathered} 28=3^{2\sin 2\alpha -1}+3^{4-2\sin 2\alpha } \left(\sin ce 2b=a+c\right) \\ \text{ Let }\mathrm{a}=3^{2\sin 2\alpha } \end{gathered}$$

$\begin{array}{c}\begin{array}{r}28=\frac{\mathrm{a}}{3}+\frac{3^4}{\mathrm{a}}\\ \to {\mathrm{a}}^2-84\mathrm{a}+243=0\to \mathrm{a}=81,3\\ \text{ If }\mathrm{a}=3\to 3^{2\sin 2\alpha }=3^1 \\ \left(if a=81 then \sin 2\alpha =2 which is not possible\right)\\ \to \sin 2\alpha =1/2\to 2\alpha =30\\ \alpha ={15}^{\circ }\end{array}\\ \text{ the first three terms are }1,14,27\\ \text{ Then }{\mathrm{t}}_5=\mathrm{a}+4\mathrm{d}\\ =1+52=53\end{array}$