Sequences
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.
Arithmetic Sequences
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. 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. Let's define a few things about arithmetic sequences:
Arithmetic sequence: a set of numbers in numerical order, with a common difference between each term.
Term (in a sequence): refers to the place or order of a number in a sequence, such as first, second, third, etc.
Common difference: the numerical distance between any two consecutive terms in an arithmetic sequence, a constant value.
Now that we know that {1, 3, 5, 7, 9...} is an arithmetic sequence with a common difference of 2, what are the values of the next two terms in the sequence? We just continue to add 2 to the last term we know. The value of the next two terms are 11, and 13.
The Formula for Finding a Term in an Arithmetic Sequence
It was easy enough to find the value of the next two terms in the sequence above, because we could just add two a few times. What if we wanted to find the value of the 50th term? Or the 400th term? Certainly adding two by hand hundreds of times isn't the easiest way. Instead, we can use this formula:
Next, we are going to use this formula to write formulas for specific arithmetic sequences, and then solve for the value of the nth term, as well as find n given its value.
Writing a Formula for an Arithmetic Sequence
Consider this sequence:
How can we write the formula to describe the value of any term in this sequence? We know that part of the formula is the value of the first term, so we know that 7 will be included in the formula. The biggest thing is to find the common difference. Remember that the common difference the the numerical distance between any two consecutive terms in an arithmetic sequence. So just pick two terms that are next to each other, and subtract one from the other. Let's choose the terms 23 and 19. 23 – 19 = 4.
It is a good idea to check that the common difference is 4 throughout the sequence (in case what we are working with isn't arithmetic at all! Also keep in mind that the common difference may be negative; this is the case when the terms are decreasing in value rather than increasing.
So we have our common difference of 4, and the initial term is 7. Let's put this into our formula:
Use the Formula to find a(n) and n
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. First, let's find the value of the 18th term.
Recall that the variable n stands for the term number. So to find the value of the 18th term, we substitute 18 in for n and solve:
Next, let's use the formula to find the term that has the value of 255. Here, we need to solve for n, given that a(n) = 255.
This means that the value of the 63rd term in the arithmetic sequence is 255.
a set of numbers in numerical order, with a common difference between each term
the numerical distance between any two consecutive terms in an arithmetic sequence; a constant value
refers to the place or order of a number in a sequence, such as first, second, third, etc.