Question

If $x_n=\cos \frac{\pi }{3^n}+i\sin \frac{\pi }{3^n}$, then $x_1.x_2.x_3....$ equal to

• A. 1
• B. -1
• C. i
• D. - i

Answer 1

Answer: (C)

i

$x_n=\cos \frac{\pi }{3^n}+i\sin \frac{\pi }{3^n}{x}_n=e^{i\frac{\pi }{3^n}}\Rightarrow x_1{x}_2{x}_3...=e^{i\frac{\pi }{3^1}}{e}^{i\frac{\pi }{3^2}}{e}^{i\frac{\pi }{3^3}}....\Rightarrow x_1{x}_2{x}_3=e^{i(\frac{\pi }{3^1}+\frac{\pi }{3^2}+\frac{\pi }{3^2}........)}$

Now $\frac{\pi }{3^1}+\frac{\pi }{3^2}+\frac{\pi }{3^2}.......$ is an infinite G.P. with $a=\frac{\pi }{3}$ and $r=\frac{1}{3}$

sum of infinte G.P $=\frac{a}{1-r}=\frac{\frac{\pi }{3^1}}{1-\frac{1}{3}}=\frac{\pi }{2}$

$\Rightarrow x_1{x}_2{x}_3.....=e^{i\left(\frac{\pi }{2}\right)}\Rightarrow x_1{x}_2{x}_3......=\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\Rightarrow x_1{x}_2{x}_3......=i$

So option $C$ is correct.