Determine the sum of the first n terms of a geometric sequence.

To determine the sum of the first n terms of a geometric sequence, use the formula Sn = a(1 - r^n)/(1 - r).

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant factor called the common ratio (r). The first term is denoted as a.

To find the sum of the first n terms of a geometric sequence, use the formula Sn = a(1 - r^n)/(1 - r). This formula can be derived by using the formula for the sum of an infinite geometric series and then taking the limit as n approaches infinity.

For example, consider the geometric sequence 2, 4, 8, 16, 32, ... with a = 2 and r = 2. To find the sum of the first 5 terms, substitute n = 5 into the formula:

Sn = 2(1 - 2^5)/(1 - 2)
= 2(-31)/(-1)
= 62

Therefore, the sum of the first 5 terms of the sequence is 62.

It is important to note that the formula for the sum of a geometric sequence only works if the common ratio is not equal to 1. If r = 1, then the sequence is simply a list of the same number repeated n times, and the sum of the first n terms is n times that number. If r is less than -1 or greater than 1, then the sum of the infinite series does not exist.