Problem 93 - College Algebra

Question

What is the difference between a geometric sequence and an infinite geometric series?

Step-by-Step Solution

Answer

The difference lies in the nature of each term: A geometric sequence is a set of numbers following a pattern, multiplied by a common ratio. An infinite geometric series is the sum of the terms in an infinite geometric sequence.

Step 1: Understanding Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the geometric sequence can be 2, 4, 8, 16, ... Here, each term is multiplied by the common ratio of 2 to get to the next term.

Step 2: Understanding Infinite Geometric Series

An infinite geometric series, on the other hand, is the sum of the terms in an infinite geometric sequence. For instance, if we were to add up the terms of the previous geometric sequence, it would look like this: 2 + 4 + 8 + 16 + ... , and so on. This differs from the sequence as we are not just listing the terms, but also adding them up.

Step 3: Difference Between The Two

The main difference between a geometric sequence and an infinite geometric series lies in their nature: A geometric sequence is a list of numbers that follows a pattern multiplied by a common ratio, while an infinite geometric series is the sum of the terms in an infinite geometric sequence. In other words, a sequence refers to the order of numbers following a specific rule (in this case multiplication by a common ratio) whereas a series involves the summing up of these numbers.