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Math Help - geometic series of n terms

  1. #1
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    geometic series of n terms

    A geometric series of n terms can be wiriiten:
    a + ar + ar^2 + ar^3 +... + ar^n-1 where a, is the first and r is the common ratio, how would you put this in sigma notation? How can a formula be written in a, r and n for the sum of this series?
    Thank you!
    Keith Stevens
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  2. #2
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    Once again it is simple: a\sum\limits_{k = 0}^{n - 1} {r^k }.
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  3. #3
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    Quote Originally Posted by kcsteven View Post
    A geometric series of n terms can be wiriiten:
    a + ar + ar^2 + ar^3 +... + ar^n-1 where a, is the first and r is the common ratio, how would you put this in sigma notation? How can a formula be written in a, r and n for the sum of this series?
    Thank you!
    Keith Stevens
    Observe that if we multiply the series by 1-r we get all the terms but the first and last cancelling:

    <br />
\left[ \sum_{k=0}^n a\,r^k \right](1-r)=a(1-r^{n+1})<br />

    so:

    <br />
\sum_{k=0}^n a\,r^k =a\,\frac{1-r^{n+1}}{1-r}<br />
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  4. #4
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    Hello, Keith!

    Here's where the summation formula comes from . . .


    We have the series: . S \;=\;a + ar + ar^2 + ar^3 + \cdots + ar^{n-1}

    . . . . Multiply by r\!:\;rS \;=\;\qquad ar + ar^2 + ar^3 + \cdots + ar^{n-1} + ar^n


    Subtract and we get: . S - rS \;=\;a - ar^n

    Therefore: . (1 - r)S \;=\;a(1- r^n)\quad\Rightarrow\quad \boxed{S \;=\;a\,\frac{1-r^n}{1 - r}}

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