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Math Help - Complex Formula Proof

  1. #1
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    Complex Formula Proof

    Suppose q = e^[(2*pi*i)/n]

    Show that:

    1 + 2*q + 3*q^2 + ... + n*q^(n-1) = n/(q-1)

    After testing with various n I have found that this is true, however I am having trouble with the general proof.

    Thanks!
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  2. #2
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    Quote Originally Posted by KatyCar View Post
    Suppose q = e^[(2*pi*i)/n]

    Show that:

    1 + 2*q + 3*q^2 + ... + n*q^(n-1) = n/(q-1)

    After testing with various n I have found that this is true, however I am having trouble with the general proof.

    Thanks!
    S_n = 1 + q + q^2 + ... + q^n is a geometric series. The sum is well known. Differentiate both sides.
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  3. #3
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    You could also note that 1- z^n= (1- z)(1+ z+ z^2+ z^3+ \cdot\cdot\cdot\+ z^{n-2}+ z^{n- 1}) to get a similar result to the "geometric series". Of course, it is recognizing that you can differentiate term by term that is the crucial step.
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