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  • If  and <img alt="3^{4-2\sin 2\alpha }" src="http://entrancecorner.codecogs.com/gif.latex?3%5E%7B4-2%5Csi

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 :
Option: 1 66
Option: 2 81
Option: 3 65
Option: 4 78

Answers (1)

best_answer

Three-term are in AP

3^{2 \sin 2 \alpha-1}, 14 \text { , } 3^{4-2 \sin \alpha}

Then

\\3^{4-\sin 2 \alpha}+3^{2 \sin 2 \alpha-1}=28 \\ \text { Let } 3^{2 \sin 2 \alpha}=\mathrm{t} \\ \frac{81}{\mathrm{t}}+\frac{\mathrm{t}}{3}=28 \\ \mathrm{t}=81,3 \\ 3^{2 \sin 2 \alpha}=3^{1}, 3^{4}

\sin 2 \alpha=\frac{1}{2}, 2(\text { rejected })

\begin{aligned} &\begin{array}{l} \text { First term } \mathrm{a}=3^{2 \sin 2 \alpha-1} \\ \mathrm{a}=1 \\ \text { Second term }=14 \end{array}\\ &\therefore \text { common difference } \mathrm{d}=13\\ &\mathrm{T}_{6}=\mathrm{a}+5 \mathrm{~d}\\ &\begin{array}{l} \mathrm{T}_{6}=1+5 \times 13 \\ \mathrm{~T}_{6}=66 \end{array} \end{aligned}

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himanshu.meshram

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