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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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    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

1 Answers 1

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First, this is not really a statistical question. You seem to confuse regression and recursion. I assume your formula should be

$ C_{i+1} = a \sum_{j=0}^{i}{C_j} + b $

and i takes on integer values with $ i\geq 0$. Let's rewrite this as

\begin{equation} C_{i+1} = a \sum_{j=0}^{i-1}{C_j} + aC_i +b \end{equation}

Note that

$ C_i = a \sum_{j=0}^{i-1}{C_j} + b. $

Subtract b from both sides of that equation and use the result to eliminate the sum in the formula above for $C_{i+1}$. That's all the help I'm willing to give, unless you show that you worked on this on your own. But be very careful when arriving at $i=1$.

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    Do the results differ when you start with$a$different value for $C_1$? If yes, then no.2012-05-09