The formula for the linear pattern of a linear equation describes the relationship between terms in a sequence where the difference between consecutive terms is constant. This is also known as an arithmetic sequence.
Understanding the Formula
A linear number pattern (arithmetic sequence) is a series of numbers where the difference between each number is constant. The general formula to find the nth term of such a sequence is given by:
an = dn - c
Where:
- an represents the nth term in the sequence.
- d is the common difference between consecutive terms.
- n is the position of the term in the sequence.
- c is a constant value.
Dissecting the Components
Let's break down each component of the formula:
- Common Difference (d): This is the constant amount added or subtracted to get from one term to the next. For example, in the sequence 2, 4, 6, 8..., the common difference is 2.
- Term Position (n): This indicates which term in the sequence you want to find. For instance, if you want to find the 5th term, n would be 5.
- Constant (c): This value helps adjust the formula to fit the specific sequence. It is derived from the first term of the sequence.
Finding the Constant c
The constant c can be determined using the first term of the sequence (a1):
c = d - a1
This constant ensures that the formula correctly calculates all terms in the sequence.
Example
Consider the linear sequence: 5, 8, 11, 14…
- Find the common difference (d): 8 - 5 = 3, therefore d = 3.
- Find the constant (c): c = d - a1 = 3 - 5 = -2.
- Write the formula: an = 3n - (-2) which simplifies to an = 3n + 2.
Now you can find any term in the sequence. For example, the 10th term (a10) is:
a10 = 3(10) + 2 = 32
Summary Table
| Variable | Description |
|---|---|
| an | The nth term of the sequence |
| d | Common difference |
| n | Position of the term in sequence |
| c | Constant Value |
