To find a specific term in a geometric sequence, you use a formula that takes into account the first term and the common ratio of the sequence.
Understanding Geometric Sequences
A geometric sequence is a list 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.
The Formula
According to the reference, the formula for a geometric sequence is:
a(r)^n-1
Where:
- a is the first term of the sequence.
- r is the common ratio (the number you multiply each term by to get the next).
- n is the position of the term you want to find.
Steps to Find a Specific Term
Here's a breakdown of how to use the formula:
- Identify the first term (a): This is the initial number in your sequence.
- Determine the common ratio (r): Divide any term by its preceding term. For example, if your sequence is 2, 4, 8, 16, the common ratio would be 4/2 = 2, or 8/4 = 2.
- Specify the term number (n): Decide which term in the sequence you are trying to calculate (e.g., the 5th term, the 10th term).
- Apply the formula: Plug the values of a, r, and n into the formula a(r)^n-1 and calculate the result.
Example
Let's say you have a geometric sequence: 3, 6, 12, 24... and you want to find the 5th term.
- a (first term) = 3
- r (common ratio) = 6 / 3 = 2
- n (term number) = 5
Applying the formula a(r)^n-1:
3 (2)^(5-1) = 3 (2)^4 = 3 * 16 = 48
Therefore, the 5th term in the sequence is 48.
Summary Table
| Element | Description | Example (using our sample sequence) |
|---|---|---|
| a | The first term of the sequence. | 3 |
| r | The common ratio (what you multiply by to get the next term). | 2 |
| n | The position of the term you are trying to find. | 5 |
By following these steps and utilizing the provided formula, you can successfully find any term in a geometric sequence.
