Suddenly, I encounter this problem: find the sum

Here is my way of thinking,

since

where

then

The sum break down to calculate

The sum is related to n-root of unity as

because

break down the product,

thus,

And since , thus, and

The graphical imagination is that, the solutions of n-root of unity are the vectors, isotropically distributed from center of a unit circle to its circumference. Thus, the add up of all these vectors will form a N-size polygon, and the result of the sum is zero.

I found it is very interesting to transform one question into another question that can be solved graphically.

This problem is originated from 3-phase AC power that . I wonder, is it the total power be a constant for , and the answer is yes.

The ultimate reason for that is, when generating AC power, the motor is rotating at constant speed and a constant power is inputting to the system, which can using 1-phase (), 2-phase (), 3-phase (), and so on, to generate the AC power. Thus, as long as the phase distributed equally, the sum of power must be a constant.

### Like this:

Like Loading...

## Leave a Reply