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Consider the following recursion: $$C_{i+1} = a \sum_{j=1}^iC_j + b$$ where $a$ and $b$ are constants.

  1. Can series-element $C_i$ be expressed in terms of only its index $i$, $a$ and $b$?
  2. In case $C_1$ = $b$, does the answer change? simplified expression?

Thanks much.

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    Sounds like homework. Please format your question better. For example Ci+1 probably is supposed to be an element of a series and would be better written down as $C_{i+1}$. Are i and j really supposed to be different? I don't think so. What values is i supposed to take? Positive integers, $i \geq 0$? I also suspect you mean recursion instead of regression, which would make this not even a a statistical question. Please clear up your question.2012-05-09
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    Ignore the question about the difference between i and j. The pertinent question is what range of j you sum over. Unfortunately I could not edit the comment anymore. @Ken: I think your edit should have $C_{i+1}$ on the left hand and not $C_{i}+1$, but until the OP states his intentions I am hesistant to edit myself.2012-05-09
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    Forgot to add brackets around it, thanks.2012-05-09
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    Really Erik essentially did the whole thing for you. The final algebraic step yields Ci+1-Ci=aCi or Ci+1 = (a+1) Ci2012-05-09
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    Assume $a=0$, $C_1=0$, $b \ne 0$. $C_2=b$, not $(a+1)C_1$, which equals $0$.2012-05-09
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    To finish what Erik said you get Ci+1=(a+1)^(i-1) C2 = (a+1)^(i-1) (aC1+b)2012-05-09
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    With over a 1000 reputation points across the network, I would expect you to be familiar with the "Edit" function. Please edit your other answer to this question instead of this one.2012-05-09

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