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Math Help - Summing a series.

  1. #1
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    Summing a series.

    Sum the series

    Sn = a + (a+d)r + (a+2d)r^2 + ... + [a+(n-1)d]r^(n-1)

    n-1
    Sum (a+nd)r^n
    n=0

    Presumably I need to split this up into a sum of ar^n and ndr^n

    I have already summed ar^n, how would I go about summing ndr^n ?
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  2. #2
    hpe
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    Quote Originally Posted by SunGod
    Sum the series

    Sn = a + (a+d)r + (a+2d)r^2 + ... + [a+(n-1)d]r^(n-1)

    n-1
    Sum (a+nd)r^n
    n=0

    Presumably I need to split this up into a sum of ar^n and ndr^n

    I have already summed ar^n, how would I go about summing ndr^n ?
    Split this up as follows:
     S_0 = a + ar + ar^2 + \dots + ar^n = a\frac{r^{n+1}-1}{r-1}
     S_1 = dr + dr^2 + \dots + dr^n = dr\frac{r^n-1}{r-1} = d \frac{r^{n+1}-r}{r-1}
     S_2 = dr^2 + \dots + dr^n =  dr^2\frac{r^{n-1}-1}{r-1} = d\frac{r^{n+1}-r^2}{r-1}
    and so on until
     S_n = dr^n = d \frac{r^{n+1}-r^n}{r-1}

    Then add all these sums, using the formula for geometric sums. There is also a way to do this with calculus.
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  3. #3
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    Hey thanks for the solution, I've already handed in the work alas without the proof of that question, but at least now I know the logic behind working out a complicated summation like that (complicated on my level, anyway)

    I've also read the other solution which requires calculus, thanks alot
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