The sequence 3, 7, 11, 15 is an arithmetic sequence. This type of sequence is characterized by a constant difference between consecutive terms.
Understanding Arithmetic Sequences
An arithmetic sequence is a progression of numbers such that the difference between the consecutive terms is constant. This constant value is known as the common difference (often denoted by d). In the sequence 3, 7, 11, 15, we can observe this pattern:
- 7 - 3 = 4
- 11 - 7 = 4
- 15 - 11 = 4
Since there is a common difference of 4 between each term, this is an arithmetic sequence.
Key Characteristics of This Sequence
To fully describe an arithmetic sequence, we identify its first term and common difference.
- First Term (a₁): The initial number in the sequence. For 3, 7, 11, 15, the first term is 3.
- Common Difference (d): The constant value added to each term to get the next term. As calculated above, the common difference is 4.
Formula for the nth Term
The general formula to find any term (an) in an arithmetic sequence is:
- an = a₁ + (n - 1)d
Where:
- an = the nth term you want to find
- a₁ = the first term
- n = the term number (e.g., 1st, 2nd, 3rd, etc.)
- d = the common difference
Example: To find the 5th term (a₅) of the sequence 3, 7, 11, 15:
- a₅ = 3 + (5 - 1) * 4
- a₅ = 3 + (4) * 4
- a₅ = 3 + 16
- a₅ = 19
So, the next term in the sequence after 15 would be 19.
Sequence Overview
This table summarizes the terms and their corresponding values for the given sequence:
| Term Number (n) | Term Value (an) | Calculation (a₁ + (n - 1)d) |
|---|---|---|
| 1 | 3 | 3 + (1 - 1) * 4 = 3 |
| 2 | 7 | 3 + (2 - 1) * 4 = 7 |
| 3 | 11 | 3 + (3 - 1) * 4 = 11 |
| 4 | 15 | 3 + (4 - 1) * 4 = 15 |
