Terms of an Arithmetic Sequence

Reference > Mathematics > Algebra > Sequences and Series

An arithmetic sequence is a sequence of numbers in which every pair of successive terms has a common difference. Here's the most simple arithmetic sequence:

1, 2, 3, 4, 5, ...

Can you see the pattern? Every number in the sequence is found by taking the previous number and adding 1 to it. Thus, 1 is the common difference.

Here's another example:

-4, -2, 0, 2, 4, ...

In this sequence each term ("term" is a fancy word for a number in a sequence) is two more than the term before it. Thus, the common difference is 2.

Suppose I wanted to find the 6^th term of the sequence above. Well, that's easy; I just add 2 to the 5^th term, and I've got 6.

Of course, if I wanted to find the 100^th term, that might be a little more time consuming! So it would be nice if we could find a formula for the n^th term. Something that looks like this:

a_n = -4 + 2(n - 1)

Does that formula work for our sequence? Sure it does! The first term is -4 + 2 x 0 = -4. You can plug in other values and see that they work.

The big question is: how did I come up with the formula?

Well, I reasoned that since the common difference was 2, if I started with the first term, I had to add the common difference (n - 1) times in my formula.

The general formula, where a₁ is the first term, andd is the common difference:

a_n = a₁ + d(n -1)