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Math Help - A Particular Summation Serie

  1. #1
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    A Particular Summation Serie

    Hello All,

    I am just wondering if this following has a formula:

    i = 0k - 1 Xi/Yi = ?

    For example, X = 3, and Y = 2, and k = 5

    30/20 + 31/21 + 32/22 + 33/23 + 34/24 = 1 + 1.5 + 2.25 + 3,375 + 5.0625 = 13.1875;

    Any hints?

    Thanks!
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  2. #2
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    Re: A Particular Summation Serie

    This is a geometric progression.
    Thanks from mohamedennahdi
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  3. #3
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    Re: A Particular Summation Serie

    Quote Originally Posted by mohamedennahdi View Post
    I am just wondering if this following has a formula:
    i = 0k - 1 Xi/Yi = ?
    For example, X = 3, and Y = 2, and k = 5
    30/20 + 31/21 + 32/22 + 33/23 + 34/24 = 1 + 1.5 + 2.25 + 3,375 + 5.0625 = 13.1875;
    S = \sum\limits_{k = o}^{N - 1} {{{\left( {\frac{x}{y}} \right)}^k}}  = 1 + \left( {\frac{x}{y}} \right) +  \cdots  + {\left( {\frac{x}{y}} \right)^{N - 1}}

    \left( {\frac{x}{y}} \right)S = \left( {\frac{x}{y}} \right) + {\left( {\frac{x}{y}} \right)^2} \cdots  + {\left( {\frac{x}{y}} \right)^N}

    \left( {1 - \frac{x}{y}} \right)S = 1 - {\left( {\frac{x}{y}} \right)^N}

    Thus S = \dfrac{{1 - {{\left( {\frac{x}{y}} \right)}^N}}}{{\left( {1 - \frac{x}{y}} \right)}}
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  4. #4
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    Re: A Particular Summation Serie

    Wow, this is really a "in a nutshell" response.
    However, I can't figure out how we moved from line 2 to line 3.
    Can you elaborate if you don't mind?

    Thanks!
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  5. #5
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    Re: A Particular Summation Serie

    The Zeno sum
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  6. #6
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    Re: A Particular Summation Serie

    Quote Originally Posted by mohamedennahdi View Post
    Wow, this is really a "in a nutshell" response.
    However, I can't figure out how we moved from line 2 to line 3.
    Can you elaborate if you don't mind?

    Thanks!
    From Line 2:

    \displaystyle \begin{align*} \left(\frac{x}{y}\right)S &= \left(\frac{x}{y}\right) + \left(\frac{x}{y}\right)^2 + \left(\frac{x}{y}\right)^3 + \dots + \left(\frac{x}{y}\right)^{N - 1} + \left(\frac{x}{y}\right)^N \\ 1 + \left(\frac{x}{y}\right)S &= 1 + \left(\frac{x}{y}\right) + \left(\frac{x}{y}\right)^2 + \left(\frac{x}{y}\right)^3 + \dots + \left(\frac{x}{y}\right)^{N - 1} + \left(\frac{x}{y}\right)^N \\ 1 + \left(\frac{x}{y}\right)S &= S + \left(\frac{x}{y}\right)^N \\ 1 - \left(\frac{x}{y}\right)^N &= S - \left(\frac{x}{y}\right)S \\ 1 - \left(\frac{x}{y}\right)^N &= S\left[ 1 - \left(\frac{x}{y}\right) \right] \\ S &= \frac{1 - \left(\frac{x}{y}\right)^N}{1 - \left(\frac{x}{y}\right)} \end{align*}
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  7. #7
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    Re: A Particular Summation Serie

    In other words, line 3 in post #3 is the difference of the first two lines.
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