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Math Help - The general arithmetic-geometric series

  1. #1
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    Smile The general arithmetic-geometric series

    I am looking for the general arithmetic-geometric series of n terms in the form of: a = (a+d)r + (a + 2d)r^2 + ...+[a + (n-1)d]r^n-1
    where the first factors of each term form an arithmetic series and the second factors from a geometric series. I am looking for a website that shows this step by step so I can be able to explain this in my education class, I am in calculus and we have not covered this material as of yet. I looked in my calculus text and the method to derive this formula is not listed. Can someone point me in the right direction.
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  2. #2
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    Hello, Keith!

    This is not a pleasant problem . . . especially without LaTeX!


    I am looking for the general arithmetic-geometric series of n terms in the form of:

    .S . = . a = (a+d)r + (a+2d)r + (a+3d)r + ... + [a + (n-1)d]r^{n-1}

    We start with:
    . . S . = . a + (a+d)r + (a+2d)r + (a+3d)r + . . . + [a + (n-1)d]r^{n-1}

    Multiply by r:
    . .rS . = . . . . . ar . + . .(a+d)r + (a+2d)r + . . . + [a + (n-2)d]^{n-1} + [a + (n-1)d]r^n

    Subtract:
    . . S - rS . = . a + dr + dr + dr + . . . + dr^{n-1} - [a + (n-1)d]r^n
    . . . . . . . . . . . . . \_________________________/


    The middle group is a geometric series with first term dr,
    . . common ratio r, and n terms.

    . . . . . . . . . . . 1 - r^n
    Its sum is: . dr ---------
    . . . . . . . . . . . .1 - r

    . . . . . . . . . . . . . . . . . . . . . . . .1 - r^n
    So we have: . (1 - r)S . = . a + dr -------- - [a + (n-1)d]r^n
    . . . . . . . . . . . . . . . . . . . . . . . . .1 - r


    I'll let you solve for S . . . I need a nap!

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  3. #3
    Junior Member
    Joined
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    Smile arithmetic-geometric series of n terms

    Soroban is this correct?
    (1-r)S = a(1-r^n)/1-r + dr(1-r^n-1)/(1-r)^2 - [a = (n-1)d]r^n/(1-r)

    Thank you,
    Keith Stevens
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