Geometric series | Mathematico

It is natural to ask, what happens when we add together the terms in a geometric sequence? In other words, we want to find a formula for

S_{n} = a + a r + a r^{2} + \dots + a r^{n - 1}

The formula needed is:

S_{n} = a \frac{1 - r^{n}}{1 - r}

As with arithmetic series, you need to know the proof for this formula. The proof is discussed in the examples and you can have a go for yourself in the problems.

Find the sum of the first $10$ terms of the geometric sequence which begins
$3, 6, 12, 24, \dots$
Find the value of
$\sum_{k = 3}^{12} 160 \times {(\frac{1}{2})}^{k}$