The ratio of a geometric sequence is found by dividing any term in the sequence by the term that precedes it. This constant ratio is often referred to as the "common ratio."
Understanding Geometric Sequences
A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant value (the common ratio). This is different from an arithmetic sequence, where a constant value is added to each term.
Finding the Common Ratio
Here's how to find the common ratio (often denoted as 'r'):
- Choose any two consecutive terms in the sequence. Let's say you have the terms an and an-1, where an is the current term and an-1 is the term immediately before it.
- Divide the current term by the previous term: r = an / an-1
- Verify: To ensure accuracy, repeat this process with a different pair of consecutive terms. The ratio should be consistent throughout the sequence.
Example
Consider the geometric sequence: 2, 6, 18, 54, ...
- Choose the terms 6 and 2.
- Divide: r = 6 / 2 = 3
- Verify with another pair: 18 and 6. r = 18 / 6 = 3
- Verify again: 54 and 18. r = 54 / 18 = 3
The common ratio is 3. Each term is obtained by multiplying the previous term by 3.
Formula
The formula to calculate the common ratio is:
r = an / an-1
Where:
- r = common ratio
- an = the nth term in the sequence
- an-1 = the (n-1)th term in the sequence (the term before an)
Importance of a Consistent Ratio
It's crucial to ensure that the ratio is consistent throughout the sequence. If the ratio varies between different pairs of consecutive terms, then the sequence is not a geometric sequence.
Summary
To find the ratio of a geometric sequence, divide any term by its preceding term. This consistent ratio defines the geometric sequence.
