Question

State whether the following statement is true or false.

If $Z_r=cos\frac{\pi }{3^r}+isin\frac{\pi }{3^r},r=1,2,3,......$ then $z_1{z}_2{z}_3........\infty =i$

• A. True
• B. False

Answer 1

The correct option is A True

$z_1=cos\frac{\pi }{3}+isin\frac{\pi }{3}=e^{i\frac{\pi }{3}}$

$z_2=cos\frac{\pi }{3^2}+isin\frac{\pi }{3^2}=e^{i\frac{\pi }{3^2}}$

$z_3=cos\frac{\pi }{3^3}+isin\frac{\pi }{3^3}=e^{i\frac{\pi }{3^3}}$

So,

$z_1{z}_2{z}_3.....\infty =e^{i\pi /3}{e}^{i\pi /3^2}{e}^{i\pi /3^3}....\infty =e^{i\pi [1/3+1/3^2+1/3^3+.....\infty ]}=e^{i\pi /2}=cos\left(\pi /2\right)+isin\left(\pi /2\right)=0+i=i$