Sum of First Four Terms of Geometric Sequence

Calculate the sum of the first four terms of the sequence given by $\frac{3}{2}^{n-1}$.

In this problem, you are tasked with calculating the sum of the first four terms of a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. Understanding the properties and formulas related to geometric sequences is essential in discrete mathematics, especially when analyzing sequences and series.

The sequence given here has terms of the form three over two raised to the power of n minus one, representing a common example of exponential growth. The solution involves identifying the first four terms of the sequence, summing them up using the formula for the sum of a finite geometric series. To solve this problem, you need to recognize that the common ratio in this sequence is a crucial element, as it dictates how each term is generated from the previous one.

Furthermore, understanding this problem provides a pathway to mastering other related concepts such as convergence of series, manipulation of exponents, and the broader applications of sequences in computer science and mathematics. As you watch the solution video, pay special attention to how each term is derived and how the series' sum formula is applied.