Table of Contents |
In math, a sequence is a set of numbers in a particular order. For example, {1, 3, 5, 7, 9} is a sequence, because we have numbers in a set (denoted by the curly braces) and the numbers are in numerical order. This particular sequence is finite because there are a limited number of terms (there are only 5 numbers).
Sequences can be infinite, meaning that they are an endless number of terms. {1, 3, 5, 7, 9...} is an infinite sequence, denoted by the ellipsis (the "dot-dot-dot"). This means that the sequence continues to include all positive odd integers.
The example sequence above also happens to be an example of an arithmetic sequence. Like all sequences, it is a set of numbers in numerical order. What makes it an arithmetic sequence is the constant change in value from one term to the next.
EXAMPLE
In the sequence {1, 3, 5, 7, 9...}, we add 2 to each term to get the value of the term after it.This constant change in value is called the common difference between terms.
It was easy enough to find the value of the next two terms in the sequence above because we could just add 2 a few times. What if we wanted to find the value of the 50th term? Or the 400th term? Certainly adding 2 by hand hundreds of times isn't the easiest way. Instead, we can use this formula:
In this formula, we can define each variable as:
To write the formula for an arithmetic sequence, we need to identify variables that are available with the given information.
EXAMPLE
Consider this sequence: {7, 11, 15, 19, 23, 27, 31, ... }Now that we have a formula to describe the arithmetic sequence above, we can use it to find the value of the nth term, as well as find the term number that has a specific value.
Let's first find the value of an nth term.
EXAMPLE
For the sequence {7, 11, 15, 19, 23, 27, 31, ... }, find the value of the 18th term using the formula .Substitute 18 in for n | |
Subtract 1 from 18 | |
Multiply 4 times 17 | |
Add 7 and 68 | |
Our solution |
Now let's find the term number that has a specific value.
EXAMPLE
For the sequence {7, 11, 15, 19, 23, 27, 31, ... }, use the formula to find the term that has the value of 255.Substitute 255 in for | |
Distribute 4 into | |
Combine 7 and -4 | |
Subtract 3 from both sides | |
Divide both sides by 4 | |
Our solution |
Suppose you opened a deposit account. In the first month, you made an initial deposit of $1800. You plan to contribute $150 a month after your initial deposit. The account does not pay any interest.
After how many months will you have a total of $3000?
The amount in your deposit account can be modeled using the formula for an arithmetic sequence, . In this formula:
Use the formula for arithmetic sequences and substitute the values: , , and | |
Distribute 150 into | |
Combine 18000 and -150 | |
Subtract 1650 from both sides | |
Divide both sides by 150 | |
Our solution |
It will take 9 months to have a total of $3,000 in your deposit account. You can use a table to validate the solution.
Month | Deposit Account Total |
---|---|
1 | $1800 |
2 | $1950 |
3 | $2100 |
4 | $2250 |
5 | $2400 |
6 | $2550 |
7 | $2700 |
8 | $2850 |
9 | $3000 |
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License