Question

In a $G.P.$ first term and common ratio both are equal to $\frac{1}{2}(\sqrt{3}+i).$ Then the modulus value of $n^{th}$ term of the $G.P.$ is

• A. $2^n$
• B. $4^n$
• C. $2^{n-1}$
• D. $1$

Answer 1

The correct option is D $1$

Let $a$ and $r$ be the first term and common ratio of the $G.P.,$ then
$a=r=\frac{1}{2}(\sqrt{3}+i)$

Now,
$T_n=a{r}^{n-1}={\left[\frac{1}{2}\right(\sqrt{3}+i\left)\right]}^n$
$={[\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}]}^n=\cos \frac{n\pi }{6}+i\sin \frac{n\pi }{6}$

$\therefore |T_n|=\sqrt{{\cos }^2\frac{n\pi }{6}+{\sin }^2\frac{n\pi }{6}}=1$