Calculating this sum in $O(n)$ time

Question

Assume that $\forall 1 \leq i \leq n : a>0, c_i > 0 $ where $n \in \mathbb{N}$. I need to show that it is possible to compute this sum in $O(n)$ time:

$$I = \sum_{(\epsilon_1,...,\epsilon_n) \in \{0,1\}^n}a^{c_1 \epsilon_1 + ... + c_n \epsilon_n}$$

(a sum over all binary vectors of length $n$) I notice that in $I$, for any fixed $1\leq i \leq n$, we have that $a^{c_i\epsilon_i} = 1$ in $2^{n-1}$ of the times, and $a^{c_i}$ in the other $2^{n-1}$ of the times. So $I$ can be written as follows:

$$2^{n-1}\cdot a^{c_1} \cdot \sum_{(\epsilon_2,...,\epsilon_n)\in \{0,1\}^{n-1}}a^{c_2\epsilon_2}\cdot ... \cdot a^{c_n \epsilon_n} + 1 \cdot \sum_{(\epsilon_2,...,\epsilon_n)\in \{0,1\}^{n-1}}a^{c_2\epsilon_2}\cdot ... \cdot a^{c_n \epsilon_n}$$

But coming to expand this for $i=2,...,n$ I immediately realize that this won't turn out to be $O(n)$. I would appreciate some guidance.