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 $i=1$.