To determine the explicit formula for an arithmetic sequence, we use the formula: an = a + (n - 1)d, where an represents the nth term of the sequence, a is the first term, n is the term number, and d is the common difference.
Understanding the Explicit Formula
The explicit formula allows us to calculate any term in the sequence directly without needing to know the previous terms. This contrasts with recursive formulas, which rely on knowing earlier terms in the sequence to calculate subsequent terms.
Components of the Formula
- an: This is the term we want to find. It's the nth term in the sequence.
- a: This represents the first term of the arithmetic sequence.
- n: This is the position of the term in the sequence. For example, if we want the 5th term, n would be 5.
- d: This is the constant difference between any two consecutive terms in the arithmetic sequence.
Applying the Formula
Let's say we have a hypothetical arithmetic sequence table where:
| n | an |
|---|---|
| 1 | 3 |
| 2 | 7 |
| 3 | 11 |
| 4 | 15 |
Here's how to determine the explicit formula:
- Identify the First Term (a): In our example, the first term (a) is 3.
- Find the Common Difference (d): The common difference is the constant amount by which each term increases. In this table, it is 7 - 3 = 4, and 11 - 7 = 4, etc. So, d = 4.
- Plug Values into the Formula: Substituting a = 3 and d = 4 into the explicit formula an = a + (n - 1)d gives us:
an = 3 + (n - 1)4 - Simplify the Formula: Distribute the 4 to get:
an = 3 + 4n - 4
an = 4n - 1
Therefore, in this example, the explicit formula for the arithmetic sequence is an = 4n - 1. This formula allows you to calculate any term in the sequence. For instance, to find the 10th term, you would substitute n = 10: a10 = 4(10) - 1 = 40 - 1 = 39.
The explicit formula, an = a + (n - 1)d is a powerful tool for understanding and working with arithmetic sequences.
