Question
Find the sum of the first 9 terms of the following sequence. Round to the nearest hundredth if
necessary.

Sum of a finite geometric series:

To find the sum of the first 9 terms of the sequence , we first identify that this is a geometric series where the first term and the common ratio .

Using the formula for the sum of a finite geometric series:

We plug in , , and :

Calculating, we have:

Thus, the sum of the first 9 terms is 3420.

This geometric series has a fascinating historical background stemming from the study of sequences and series in mathematics. The general understanding of geometric progressions dates back to ancient civilizations, including the Greeks and Indians, who explored these concepts in their quest to understand ratios and proportions. Heroes during these times noted patterns in numbers that led to foundational principles in mathematics.

In real-world applications, geometric sequences are prominently used in finance, particularly in compound interest calculations. For example, if you invest an amount at a certain interest rate, the growth of your investment can be modeled using geometric series. By learning how to work with these sequences, you can make informed decisions about savings and investments, harnessing the power of exponential growth to your advantage!