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 Minimum integral value of S is equal to

• A. $1$
• B. $2$
• C. $4$
• D. $24$

Answer 1

The correct option is A $1$

Clearly $S$ is summation of infinite G.P. with common ratio $r=2{\sin }^2\theta$

For $S$ to converge, we must have $2{\sin }^2\theta <1$

$\Rightarrow S=\frac{1}{1-2{\sin }^2\theta }$
Now for minimum value of $S,1-2{\sin }^2\theta =$ maximum

$\Rightarrow 2{\sin }^2\theta =$ minimum

$\therefore \sin \theta =0$

Hence $S_{min}=1$