General formula for this sum $\sum_{i=0}^n \frac{7+2(-\frac{1}{2})^i}{3}$ and strong induction

Question

I need help to figure out the general formula of this sum:

$$\sum_{i=0}^n \frac{7+2(-\frac{1}{2})^i}{3}$$

Then prove it with strong induction.

Answer 1

Here's a start. It's usually good to split a complicated problem into simpler parts.

$\begin{array}\\ \sum_{i=0}^n \frac{7+2(-\frac{1}{2})^i}{3} &=\frac13\sum_{i=0}^n (7+2(-\frac{1}{2})^i)\\ &=\frac13\left(\sum_{i=0}^n 7+\sum_{i=0}^n2(-\frac{1}{2})^i)\right)\\ &=\frac13\left(7(n+1)+2\sum_{i=0}^n(-\frac{1}{2})^i)\right)\\ \end{array} $

Your turn.