I stumbled upon a question in a DSP course (coursera) which I don't understand, shown below is a screenshot of the question and answer.
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The part which I don't understand is circled. Why is it equal to 0? For multiples of N I understand that each element in the summation is 1 because the exponent is 0. But why does the summation equal 0 when (i+n) is not a multiple of N?


Just use the formula for the geometric sum:

$$\sum_{k=0}^{N-1}e^{-j\frac{2\pi}{N}kl}=\frac{1-e^{-j2\pi l}}{1-e^{-j\frac{2\pi}{N}l}},\qquad l\neq mN,\;m\in\mathbb{Z}$$

Now realize that $e^{-j2\pi l}=1$ for $l\in\mathbb{Z}$ and the result follows.

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