How Do You Find the Ratio of a Geometric Sequence?

To find the common ratio of a geometric sequence, divide any term by the term that precedes it.

Here's a more detailed explanation:

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant. This constant is called the common ratio, usually denoted by 'r'.

To calculate 'r', you can use the following formula:

r = a_n / a_n-1

Where:

a_n is any term in the sequence.
a_n-1 is the term immediately before a_n.

Steps to Find the Common Ratio:

Choose any term in the sequence (except the first term, as it has no preceding term).
Divide that term by the term that comes directly before it.
Verify the ratio by repeating the process with other consecutive terms. The ratio should be the same throughout the sequence.

Example:

Consider the geometric sequence: 5, 8, 11,...

To find if the sequence is geometric, you would do the following according to the transcript:

8 - 5 = 3
11 - 8 = 3

Because the difference between consecutive terms is the same, the sequence is arithmetic, not geometric. In order to determine the ratio of a geometric sequence, we must divide, rather than subtract:

Consider the geometric sequence: 2, 6, 18, 54, ...

Choose the second term, 6.
Divide it by the preceding term, 2: 6 / 2 = 3
Verify: 18 / 6 = 3, and 54 / 18 = 3

Therefore, the common ratio (r) of this geometric sequence is 3.