Arithmetic Sequences
A sequence is a set of numbers in a specific order. What this means is that the set of numbers can be put into a one-to-one correspondence with the Counting Numbers (1, 2, 3, 4, ... ). Thus, you can talk about the 1st element (or term) in a sequence or the 10th element in a sequence or the 101st element in a sequence.
An arithmetic sequence is a sequence in which the difference between any two consecutive terms is the same, i.e., the difference is a constant.
Arithmetic sequences frequently occur in problem solving situations.
Examples
The sequence that begins 1, 4, 7, 10, 13, 16, . . . is an arithmetic sequence since the difference between consecutive terms is always 3.
The sequence that begins 8, 6, 4, 2, 0, -2, -4, . . . is an arithmetic sequence since the difference between consecutive terms is always -2.
In order to identify if a pattern is an arithmetic sequence you must examine consecutive terms. If all consecutive terms have a common difference you can conclude that the sequence is arithmetic.
Consider the sequence:
4, 11, 18, 25, 32, . . .
Since:11 - 4 = 7
18 - 11 = 7
25 - 18 = 7
32 - 25 = 7
the sequence is arithmetic. We can continue to find subsequent terms by adding 7. Therefore, the sequence continues:
39, 46, 53, etc.The Formula for the nth Term in an Arithmetic Sequence
Consider the following table and look for a pattern:
| Term 1 | 4 | ||
| Term 2 | 4 + 7 | = 4 + (1 X 7) | = 11 |
| Term 3 | 11 + 7 | = 4 + 7 + 7 = 4 + (2 X 7) | = 18 |
| Term 4 | 18 + 7 | = 4 + 7 + 7 + 7 = 4 + (3 X 7) | = 25 |
| . . . |
. . . |
. . . |
. . . |
|---|---|---|---|
| Term 12 | 4 + (11 X 7) | = 81 | |
| . . . |
. . . |
. . . |
. . . |
| Term n | = 4 + [(n - 1) X 7] |
The same strategy can be used with any arithmetic sequence.
If the first term is designated by the letter a, and the common difference is designated by the letter d,The nth term of an arithmetic sequence
= a + [(n - 1) X d]Using The Formula
Use the formula to find the 8th term of the sequence that begins with 11 and has a common difference of 4.
In this example, a = 11 (the first term), d = 4 (the common difference), and n = 8 (the term we are looking for).You can calculate the 8th term using the formula:Term 8 = 11 + [(8 - 1) X 4] = 11 + [7 X 4] = 11 + 28 = 39Try to solve a problem using what you have learned about Arithmetic Sequences.
email the author: Bruce Jacobs
Last modified: Thursday, February 6, 2003