How to guess that $ \sum_{i=1}^{n}3^i = \frac{3}{2}(3^n - 1)$ As in title how do you guess that $ \sum_{i=1}^{n}3^i = \frac{3}{2}(3^n - 1)$?
I have homework about solving recurrence relations and using iterate method I can find that http://www.wolframalpha.com/input/?i=RSolve%5B%7Ba%5Bn%5D+%3D+a%5Bn-1%5D+%2B+3%5En%2C+a%5B1%5D+%3D+3%7D%2Ca%5Bn%5D%2Cn%5D series shown in this example is equal to $ \sum_{i=1}^{n}3^i$, but I am curious how you change it into  $ \frac{3}{2}(3^n - 1)$. I know how to prove it's true, but I don't know how to guess it on my own. Anyone could explain?

Hint: $$3(3^1 + \ldots 3^k) = (3^1 + \ldots 3^k) + 3(3^{k}-1).$$
