NIMCET 2023 — Mathematics PYQ

The sum we need to find is:

Let $\alpha =e^{i\frac{2\pi }{n}}$. Then the sum becomes:

Step 2: Recognize the Geometric Progression (GP)

This is a geometric series where:

• First term ($a$) = $\alpha$

• Common ratio ($r$) = $\alpha$

• Number of terms = $n$

The formula for the sum of a GP is $S_n=\frac{a(r^n-1)}{r-1}$.

Step 3: Calculate the Sum

Substitute ${\alpha }^n$ back into the equation:

Using Euler's identity, $e^{i2\pi }=\cos (2\pi )+i\sin (2\pi )=1+0i=1$.

Now, substitute ${\alpha }^n=1$ into the sum formula:

Conclusion:

The sum of the $n^{th}$ roots of unity is always zero because the vectors representing these roots are symmetrically distributed around the origin, and their resultant sum is null.

Correct Option: (B) 0