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Math Help - finite summation formula

  1. #1
    jut
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    finite summation formula

    \sum_{k=1}^n a^k b^k=?
    Last edited by jut; September 19th 2009 at 06:32 PM.
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    Hello, jut!

    Simplify: . \sum_{k=1}^n a^k b^k

    We have: . S \;=\;ab + a^2b^2 + a^3b^3 + \hdots + a^nb^n

    This is a geometric series with: . \begin{Bmatrix}\text{first term: }ab \\ \text{common ratio: }ab \\ n\text{ terms} \end{Bmatrix}

    Its sum is: . S \;=\;ab\,\frac{(ab)^n - 1}{ab-1}

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  3. #3
    jut
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    Wow, nice Soroban!
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  4. #4
    jut
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    Quote Originally Posted by Soroban View Post
    Hello, jut!

    Its sum is: . S \;=\;ab\,\frac{(ab)^n - 1}{ab-1}

    [/size]

    One more question:

    if a=e, and b=-2, I could plug that into the above result and it would be correct?
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  5. #5
    Senior Member bkarpuz's Avatar
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    Quote Originally Posted by jut View Post
    One more question:

    if a=\mathrm{e}, and b=-2, I could plug that into the above result and it would be correct?
    sure, as long as ab\neq1, because it is a finite sum!
    In the case ab=1, you get \sum_{k=1}^{n}ab=n.
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