Writing a rule for an arithmetic sequence means defining a formula that allows you to find any term in the sequence without having to list out all the preceding terms. This formula is based on the sequence's consistent pattern.
Understanding Arithmetic Sequences
An arithmetic sequence is a list of numbers where the difference between consecutive terms remains constant. This constant difference is called the common difference (d).
For example:
- 2, 5, 8, 11, 14... (Common difference: 3)
- 10, 7, 4, 1, -2... (Common difference: -3)
The Explicit Formula
The rule for an arithmetic sequence is most often expressed using an explicit formula. This formula directly calculates the nth term (an) of the sequence, given the first term (a1), the common difference (d), and the term's position (n).
The formula is:
an = a1 + d(n - 1)
Where:
- an: The nth term in the sequence.
- a1: The first term in the sequence.
- d: The common difference between consecutive terms.
- n: The position of the term in the sequence (1st, 2nd, 3rd, etc.).
Examples
Example 1:
Let's find the 10th term (a10) of the sequence 2, 5, 8, 11, 14...
- a1 = 2
- d = 3 (5 - 2 = 3)
- n = 10
Using the formula: a10 = 2 + 3(10 - 1) = 2 + 3(9) = 29
Therefore, the 10th term is 29.
Example 2:
Find the rule for the arithmetic sequence 10, 7, 4, 1, -2...
- a1 = 10
- d = -3 (7 - 10 = -3)
The rule is: an = 10 + (-3)(n - 1) which simplifies to an = 13 - 3n
Practical Applications
The ability to write a rule for an arithmetic sequence is crucial in various fields including:
- Finance: Calculating compound interest or loan repayments.
- Physics: Modeling linear motion or other linear relationships.
- Computer Science: Generating sequences of numbers for algorithms.
