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:

• A. $65$
• B. $81$
• C. $78$
• D. $66$

Answer 1

The correct option is D $66$

$28=3^{2\sin 2\alpha -1}+3^{4-2\sin 2\alpha }$
$\Rightarrow 28=\frac{9^{\sin 2\alpha }}{3}+\frac{81}{9^{\sin 2\alpha }}$
Let $9^{\sin 2\alpha }=t$
$\Rightarrow 28=\frac{t}{3}+\frac{81}{t}$
$\Rightarrow t^2-84t+243=0$
$\Rightarrow t^2-81t-3t+243=0$
$\Rightarrow t(t-81)-3(t-81)=0$
$\Rightarrow (t-81)(t-3)=0$
$\Rightarrow t=81,3$
$\Rightarrow 9^{\sin 2\alpha }=9^2$ or $3$
$\Rightarrow \sin 2\alpha =\frac{1}{2},2\text{(not possible)}$
First term $a=3^{2\sin 2\alpha -1}$
$a=1$
Second term $=14$
$\therefore$ Common difference $d=13$
$T_6=a+5d=1+5\times 13=66$