ARITHMETIC SEQUENCE
Sequence
A Sequence is a set of things (usually numbers) that are in order.
Each number in the sequence is called a term (or sometimes "element" or "member"), read Sequences and Series for more details.
Arithmetic Sequence
In an Arithmetic Sequence the difference between one term and the next is a constant.
In other words, we just add the same value each time ... infinitely.
Example:
| 1, 4, 7, 10, 13, 16, 19, 22, 25, ... |
This sequence has a difference of 3 between each number.
The pattern is continued by adding 3 to the last number each time, like this:
In General we could write an arithmetic sequence like this:
{a, a+d, a+2d, a+3d, ... }
where:
- a is the first term, and
- d is the difference between the terms (called the "common difference")
Example: (continued)
| 1, 4, 7, 10, 13, 16, 19, 22, 25, ... |
Has:
- a = 1 (the first term)
- d = 3 (the "common difference" between terms)
And we get:
{a, a+d, a+2d, a+3d, ... }
{1, 1+3, 1+2×3, 1+3×3, ... }
{1, 4, 7, 10, ... }
Rule
We can write an Arithmetic Sequence as a rule:
xn = a + d(n−1)
(We use "n−1" because d is not used in the 1st term).
Example: Write a rule, and calculate the 9th term, for this Arithmetic Sequence:
| 3, 8, 13, 18, 23, 28, 33, 38, ... |
This sequence has a difference of 5 between each number.
The values of a and d are:
- a = 3 (the first term)
- d = 5 (the "common difference")
Using the Arithmetic Sequence rule:
xn = a + d(n−1)
= 3 + 5(n−1)
= 3 + 5n − 5
= 5n − 2
So the 9th term is:
x9 = 5×9 − 2
= 43
1. The first term of an arithmetic sequence is 4 and the tenth term is 67.
What is the common difference?
2. What is the thirty-second term of the arithmetic sequence -12, -7, -2, 3, ... ?
3. What is the fiftieth term of the arithmetic sequence 3, 7, 11, 15, ... ?