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8. Sequences and Series
hard
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
$66$
B
$65$
C
$81$
D
$78$
(JEE MAIN-2020)
Solution
Given that
$3^{4-\sin 2 \alpha}+3^{2 \sin 2 \alpha-1}=28$
Let $3^{2} \sin 2 \alpha=t$
$\frac{81}{t}+\frac{t}{3}=28$
$t=81,3$
$3^{2 \sin 2 \alpha}=3^{1}, 3^{4}$
$2 \sin 2 \alpha=1,4$
$\sin 2 \alpha=\frac{1}{2}, 2($ rejected $)$
First term $a=3^{2} \sin 2 \alpha-1$
$a=1$
Second term $=14$
$\therefore$ common difference $d=13$
$T_{6}=a+5 d$
$T _{6}=1+5 \times 13$
$T_{6}=66$
Standard 11
Mathematics
