Question

If $x _{ r }=\cos \frac{\pi}{4^{r}}+ i \sin \frac{\pi}{4^{r}}, y _{ r }= CiS \left(\frac{\pi}{3^{r}}\right)$ and $z _{ r }=x_{r} \cdot\left(y_{r}\right)^{4}$, then $z _{1} z _{2} z _{3} \ldots \ldots \ldots \ldots \ldots$..... $\infty$ is

• A. CiS $\frac{5 \pi}{6}$
• B. $-1$
• C. CiS $\frac{\pi}{3}$
• D. $i$

Answer 1

Answer: (C)

$z_{1} z_{2} z_{3} \ldots \ldots \ldots \ldots \ldots . . .$ to $\infty=\left(x_{1} x_{2} x_{3} \ldots \ldots \ldots \ldots \ldots . . . . \ldots \infty\right)\left(y_{1} y_{2} y_{3} \ldots \ldots \ldots \ldots \ldots \ldots \text {.......... } \infty\right)^{4} \left(\right.$ CiS $\left(\frac{\pi}{4}\right)$
$\operatorname{CiS}\left(\frac{\pi}{4^{2}}\right) \operatorname{CiS}\left(\frac{\pi}{4^{3}}\right)$..........to $\left.\infty\right)\left(\operatorname{CiS}\left(\frac{\pi}{3}\right) \operatorname{CiS}\left(\frac{\pi}{3^{2}}\right) \operatorname{CiS}\left(\frac{\pi}{3^{3}}\right) \ldots \ldots \ldots . . \text { to } \infty\right)^{4}$
$=\operatorname{CiS}\left(\frac{\pi}{4}+\frac{\pi}{4^{2}}+\frac{\pi}{4^{3}}+\ldots \ldots \ldots \ldots \ldots \ldots . . t o \infty\right)\left[\text { cis }\left(\frac{\pi}{3}+\frac{\pi}{3^{2}}+\frac{\pi}{3^{3}}+\ldots \ldots \ldots \ldots \ldots \ldots . . t o \infty\right)\right]^{4}$
$=\operatorname{cis} \pi\left(\frac{\frac{1}{4}}{1-\frac{1}{4}}\right)\left[\operatorname{cis} \pi\left(\frac{\frac{1}{3}}{1-\frac{1}{3}}\right)\right]^{4}=\operatorname{CiS}\left(\frac{\pi}{3}\right) i ^{4}=\operatorname{CiS}\left(\frac{\pi}{3}\right)$