- Home
- Standard 11
- Mathematics
8. Sequences and Series
normal
If ${x_r} = \cos (\pi /{3^r}) - i\sin (\pi /{3^r}),$ (where $i = \sqrt{-1}),$ then value of $x_1.x_2.x_3......\infty ,$ is :-
A
$1$
B
$-1$
C
$-i$
D
$i$
Solution
${{\rm{x}}_1} \cdot {{\rm{x}}_2} \cdot {{\rm{x}}_3}…………….\infty $
$ \Rightarrow \left( {\cos \frac{\pi }{3} – i\sin \frac{\pi }{3}} \right)\left( {\cos \frac{\pi }{9} – i\sin \frac{\pi }{9}} \right)………\infty $
$\Rightarrow \cos \left(\frac{\pi}{3}+\frac{\pi}{9}+\frac{\pi}{27}+\cdots\right)-i \sin \left(\frac{\pi}{3}+\frac{\pi}{9}+\frac{\pi}{27}+\cdots \infty\right)$
$\Rightarrow \cos \frac{\pi}{2}-i \sin \frac{\pi}{2}=-i$
Standard 11
Mathematics
