A geometric progression of form:

Can be summed from *N*_{0} to *N*_{1} inclusive as follows:

So..

Subtracting the second equation from the first yields:

or..

In particular, if N_{0}=0, N_{1}=N-1, this reduces to:

If *a* is of form:

then we have:

If we look at the pure real (fraction) in the above expression, we can observe that the numerator is zero for any integral value of *x*. The denominator is zero for any *x* which is an integral multiple of *N*. In the cases where *x* is an integral multiple of *N*, both the numerator and denominator are zero, but application of L'Hopitals rule tells us that the fraction has magnitude *N*. **The fraction is zero for all other integral x.**

So, to recap, as far as integral values of *x* are concerned, the sum is zero unless *x* is also multiple of *N* (including zero), in which case the magnitude of the sum is *N*. In fact, by considering distinct cases of *N* odd and *N* even, it is easy to show than the sum is always +*N* if *x* is a multiple of *N*. (This is not altogether surprising if we recall that original form of the expression as a sum of complex exponentials.)

In Annex A, we were presented with an expression of form:

This can be related to our result by associating..

Now both *m* and *n* range over 0..*N*-1, so..

Therefore, we need not consider any other integral multiples of *N* other than *x*=0, (*n*=*m*).

__Conclusion:__

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