How do you find the common ratio of the following geometric sequence?

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

Here's a more detailed explanation:

Understanding Geometric Sequences and the Common Ratio

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

Method for Finding the Common Ratio

1. Identify any two consecutive terms in the geometric sequence. Let's say you have the terms a_n and a_n+1, where a_n+1 immediately follows a_n.

2. Divide the second term by the first term. The common ratio, r, is calculated as:

   r = a_n+1 / a_n

3. Verify (Optional but Recommended): To ensure the sequence is indeed geometric and that you've calculated 'r' correctly, repeat the process with another pair of consecutive terms. The result should be the same.

Example

Let's say you have the geometric sequence: 2, 6, 18, 54, ...

1. Choose consecutive terms: Let's pick 6 and 2.

2. Divide: r = 6 / 2 = 3

3. Verify: Let's try another pair, 18 and 6. r = 18 / 6 = 3. The common ratio is consistent.

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

Formula Summary

Term	Description
r	Common ratio
an+1	Any term in the sequence
an	The term immediately preceding an+1
Formula	r = an+1 / an