Question

Consider the geometric progression $S=1+2{\sin }^2\theta +4{\sin }^4\theta +8{\sin }^6\theta +.....$ up to infinite terms, where S is a finite number and $\theta \ne \frac{n\pi }{2}$ where n $\epsilon$ I.
Then Values of $\theta$ always lies in the interval ?

• A. $(-\frac{\pi }{6},\frac{\pi }{6})$
• B. $(0,\frac{\pi }{3})$
• C. $(-\frac{\pi }{3},0)$
• D. $(-\frac{\pi }{4},\frac{\pi }{4})-\left\{0\right\}$

Answer 1

The correct option is D $(-\frac{\pi }{4},\frac{\pi }{4})-\left\{0\right\}$

Given $S=1+2si{n}^2\theta +4si{n}^4\theta +8si{n}^6\theta +.....$ is a GP.

Common ratio $r=2si{n}^2\theta$

Now, since $0<2si{n}^2\theta <1$

$0<si{n}^2\theta <\frac{1}{2}$

$\Rightarrow \theta \epsilon (-\frac{\pi }{4},\frac{\pi }{4})-\left\{0\right\}$