Question

What is the defining characteristic of a geometric series? Give an example.

Short answer

Short Answer: A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant value called the common ratio. The general form is a, ar, ar^2, ar^3, ..., where 'a' is the first term and 'r' is the common ratio. For example, a geometric series with a first term of 2 and a common ratio of 3 would be: 2, 6, 18, 54, 162, ...

Step-by-step solution

Definition of Geometric Series

A geometric series is a sequence of numbers in which the quotient, also known as the common ratio, of any two consecutive terms is constant. In other words, every term after the first one is found by multiplying the previous term by the same constant value.

General Form

The general form of a geometric series is given by: a, ar, ar^2, ar^3, ... , where 'a' is the first term, 'r' is the common ratio, and each term is found by multiplying the previous term by 'r'.

Example of Geometric Series

Let's create a geometric series with a first term 'a' of 2 and a common ratio 'r' of 3. We will show the first 5 terms: 1st term, a: 2 2nd term, ar: 2 * 3 = 6 3rd term, ar^2: 6 * 3 = 18 4th term, ar^3: 18 * 3 = 54 5th term, ar^4: 54 * 3 = 162 Hence, the example of a geometric series is: 2, 6, 18, 54, 162, ...