Question

The numbers $3^{2sin2\theta -1},14,3^{4-2sin2\theta }$ form the first three terms of an A.P. Its fifth term is equal to-

• A. $-25$
• B. $-12$
• C. $40$
• D. $53$

Answer 1

The correct option is D $53$

Since the numbers are in A.P.
$\therefore 28=3^{2sin2\theta -1}+3^{4-2sin2\theta }$
$\Rightarrow 28=\frac{9^{sin2\theta }}{3}+\frac{81}{9^{sin2\theta }}$
Let $x=9^{sin2\theta }$
Hence, $x^2-84x+243=0$
$\Rightarrow (x-81)(x-3)=0\therefore x=81$ or 3
$\therefore x=9^{sin2\theta }=81,3$ or $9^2,9^{1/2}$
$\therefore sin2\theta =2$ or 1/2
Since $sin2\theta$ cannot be greater than 1 so we choose $sin2\theta =\frac{1}{2}$
Hence the terms in A.P. are
$3^0,14,27$ i.e. 1, 14, 27.
$\therefore T_5=a+4d=1+4\times 13=53$