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
terms of this A.P. is:
Solutions · 1
Official solution
Given that
\[{3^{4 - \sin 2\alpha }} + {3^{2\sin 2\alpha - 1}} = 28\]
Let \[{3^{2\sin 2\alpha }}\] = t
\[ \Rightarrow \] \[{{81} \over t} + {t \over 3} = 28\]
\[ \Rightarrow \]t = 81, 3
\[ \therefore \] \[{3^{2\sin 2\alpha }}\] = 31, 34
\[\sin 2\alpha = {1 \over 2}\], 2 (rejected)
First term a = \[{3^{2\sin 2\alpha -1}}\] = 30
\[ \Rightarrow \] a = 1
Given Second term = 14
\[ \therefore \] Common difference d = 13
T6=a+5d
\[{T_6} = 1 + 5 \times 13\]
T6=66