Solveeit Logo
Login

Question

Question: If z<sub>r</sub> = cos (p/3<sup>r</sup>) + i sin (p/3<sup>r</sup>), r = 1, 2, ……, then z<sub>1</sub>...

If zr = cos (p/3r) + i sin (p/3r), r = 1, 2, ……, then z1. z2. z3. ……… =

A

1

B

– I

C

I

D

– 1

Answer

I

Explanation

Solution

Sol. zr = Cis (π3r)\left( \frac{\pi}{3^{r}} \right) = ei(π3r)e^{i\left( \frac{\pi}{3^{r}} \right)}

z1. z2. z3 ……  = ei(π3)e^{i\left( \frac{\pi}{3} \right)}. eiπ32e^{i\frac{\pi}{3^{2}}}.eiπ33e^{i\frac{\pi}{3^{3}}} ……… 

= ei(π3+π32+π33+......∞)e^{i\left( \frac{\pi}{3} + \frac{\pi}{3^{2}} + \frac{\pi}{3^{3}} + ......\infty \right)} = eiπ(131–13)e^{i\pi\left( \frac{\frac{1}{3}}{1–\frac{1}{3}} \right)}

= eiπ×12e^{i\pi \times \frac{1}{2}} = Cis π2\frac{\pi}{2}= cos π2\frac{\pi}{2} + i sin π2\frac{\pi}{2} = i