Sum of this Geometric Series $frac{1}{4}, frac{1}{12}, frac{1}{36}, frac{1}{108}, ldots, frac{1}{2916}$

$$\frac{1}{4}, \frac{1}{12}, \frac{1}{36}, \frac{1}{108}, \ldots,\frac{1}{2916}$$

I am trying to calculate the sum of this geometric series. Here's what I've got so far.

$a = \frac{1}{4}, r = \frac{1}{3}$ and $n = 7$

So $\sum_{n=7}\frac{1}{4}(\frac{1}{3})^{n}$

Somehow equals $\frac{1093}{2916}$ according to my book?

Here are my questions:

1. How do I get the sum of this geometric series


2. Does my work look correct?


3. How can I find a way to calculate the number of terms in the series that isn't "brute forcing"? This is quite inelegant.

Thanks!

Answer

The partial sum of a geometric series is

$$\sum_{k=0}^{n-1}ar^k = a\frac{1-r^n}{1-r}$$

so take $a=1/4, r=1/3, n=7$ to get

$$S = \left(\frac{1}{4}\right)\frac{1-(1/3)^7}{1-(1/3)} = \frac{1093}{2916}.$$