The general rule for the sequence 1, 5, 9, 13 is an = 4n - 3. This rule allows you to find any term (an) in the sequence by simply knowing its position (n).
Understanding Arithmetic Sequences
The sequence 1, 5, 9, 13 is an example of an arithmetic sequence. In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is known as the common difference.
To identify the common difference in this sequence:
- 5 - 1 = 4
- 9 - 5 = 4
- 13 - 9 = 4
Since the common difference is 4, we know that the general rule will involve 4n.
Deriving the General Rule (nth Term)
The formula for the nth term of an arithmetic sequence is typically given by:
an = a1 + (n - 1)d
Where:
- an is the nth term
- a1 is the first term
- n is the term number
- d is the common difference
Let's apply this to our sequence:
- First Term (a1): 1
- Common Difference (d): 4
Substitute these values into the formula:
an = 1 + (n - 1)4
an = 1 + 4n - 4
an = 4n - 3
This confirms the rule an = 4n - 3.
How the Rule Works
Let's verify the rule by calculating the first few terms:
| Term Number (n) | Calculation (4n - 3) | Term Value (an) |
|---|---|---|
| 1 | 4(1) - 3 = 4 - 3 | 1 |
| 2 | 4(2) - 3 = 8 - 3 | 5 |
| 3 | 4(3) - 3 = 12 - 3 | 9 |
| 4 | 4(4) - 3 = 16 - 3 | 13 |
As you can see, the rule accurately generates the terms of the sequence. For example, if you wanted to find the 10th term in this sequence, you would simply substitute n = 10 into the rule:
a10 = 4(10) - 3 = 40 - 3 = 37
This approach is fundamental in algebra and sequence analysis, providing a concise way to describe and extend patterns.
