Now we simplify. \(a_0 = 1\text{,}\) so we have \(3^n + \langle\text{stuff}\rangle\text{.}\) Note that all the other terms have a 2 in them. In fact, we have a geometric sum with first term \(2\) and common ratio \(3\text{.}\) We have seen how to simplify \(2 + 2\cdot 3 + 2 \cdot 3^2 + \cdots + 2\cdot 3^{n-1}\text{.}\) We get \(\frac{2-2\cdot 3^n}{-2}\) which simplifies to \(3^n - 1\text{.}\) Putting this together with the first \(3^n\) term gives our closed formula: