A constant common ratio is the unchanging factor between consecutive terms in a geometric sequence. In simpler terms, it's the value you multiply by to get from one term to the next in the sequence.
Understanding Common Ratio
The constant common ratio is a fundamental concept in geometric sequences. According to the reference, it's "the constant ratio between two consecutive terms." This means that if you divide any term in the sequence by the term that precedes it, you'll always get the same value – the common ratio.
How to Find the Common Ratio
You can easily calculate the common ratio (often denoted as 'r') using the following method:
- Select any two consecutive terms in the geometric sequence.
- Divide the later term by the earlier term.
- The result is the common ratio.
For example, if you have a sequence where the terms are 2, 4, 8, 16..., the common ratio would be 4/2 = 8/4 = 16/8 = 2.
Why is it Constant?
The "constant" part is crucial. If the ratio between consecutive terms changes, then the sequence isn't a geometric sequence, and there isn't a common ratio. The sequence must consistently multiply by the same factor for it to qualify.
Example of a Geometric Sequence and Common Ratio
| Term Number | Term Value | Calculation |
|---|---|---|
| 1 | 3 | |
| 2 | 6 | 6 / 3 = 2 |
| 3 | 12 | 12 / 6 = 2 |
| 4 | 24 | 24 / 12 = 2 |
| 5 | 48 | 48 / 24 = 2 |
In this sequence (3, 6, 12, 24, 48...), the common ratio is 2 because each term is obtained by multiplying the previous term by 2.
Use of Common Ratio
As the reference indicates, "The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly." This allows us to determine any term in the sequence if we know the first term and the common ratio.
