Expressing Series-Element in Terms of its Index

Question

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.

Answer 1

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 $i=1$.