The question is slightly misstated. Geometric Progressions (GPs) do not have common differences, they have common ratios. Here's how to find the common ratio in a Geometric Progression.
Understanding Geometric Progression (GP)
A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
How to Find the Common Ratio
The common ratio (often denoted as r) is the constant value that you multiply by each term to get the next term in the sequence. To find it, follow this simple rule:
- Divide any term by its preceding term.
Step-by-Step Calculation
- Identify two consecutive terms in the geometric progression. Let’s call the term later in the sequence an and its immediate predecessor an-1.
- Divide the later term by its preceding term. This means calculating an / an-1.
- The result is the common ratio (r).
- Repeat this for other pairs of consecutive terms to confirm the common ratio. If the ratio is consistent between different consecutive terms, then it confirms you have a GP.
Example
Let's take the GP: 2, 4, 8, 16, 32...
- First ratio: 4 / 2 = 2
- Second ratio: 8 / 4 = 2
- Third ratio: 16 / 8 = 2
- Fourth ratio: 32 / 16 = 2
As you can see, the common ratio r is consistently 2.
Using the Formula
The formula to find the common ratio is:
r = an / an-1
Where:
- r is the common ratio
- an is any term in the GP
- an-1 is the term preceding an
Practical Insights
- Consistency is Key: If the ratio is not consistent between consecutive terms, then the sequence is not a Geometric Progression.
- Can be a Fraction: The common ratio can be a fraction. For example, in the sequence 100, 50, 25,... the common ratio is 1/2 or 0.5.
- Can be Negative: The common ratio can also be negative, creating an alternating GP. For instance, the sequence 2, -4, 8, -16,... has a common ratio of -2.
Summary
| Step | Description |
|---|---|
| 1 | Choose any two consecutive terms in your Geometric Progression |
| 2 | Divide the later term by its preceding term. |
| 3 | Result is the common ratio (r). |
| 4 | Confirm the ratio with other consecutive pairs. |
By following these steps, you can effectively find the common ratio in any Geometric Progression.
