\( \sum _ { r = 0 } ^ { m } 72 ( \frac { 1 } { 3 } ) ^ { r - 1 } = \frac { 2912 } { 9 } \)

Did you know that series like this one can be tied back to concepts in ancient mathematics? The Greeks studied geometric series long before the advent of calculus, as they explored the nature of infinite sums and ratios. They laid the groundwork for our understanding of sequences and series, which are fundamental in many areas of modern mathematics, including computer science and finance. Now, let’s get practical! If you ever encounter a geometric series, remember the formula for the sum \( S_n = a \frac{1 - r^n}{1 - r} \) where \( a \) is the first term and \( r \) is the common ratio. A common mistake is forgetting to adjust for the index when applying the formula. Make sure you accurately identify your first term and the number of terms included in your series to avoid those pesky calculation errors!