How to Solve an Arithmetic Sequence Problem With Variable Terms

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A mathematical sequence is any set of numbers that are arranged in order. An example would be 3, 6, 9, 12, . . . Another example would be 1, 3, 9, 27, 81, . . . The three dots signify that the set continues. Each number in the set is called a term. An arithmetic sequence is one in which each term is separated from the one before it by a constant that you add to each term. In the first example, the constant is 3; you add 3 to each term to get the next term. The second sequence isn't arithmetic because you can't apply this rule to get the terms; the numbers appear to be separated by 3, but in this case, each number is multiplied by 3, making the difference (i.e., what you'd get if you subtracted terms from each other) much more than 3.

It's easy to figure out an arithmetic sequence when it's only a few terms long, but what if it has thousands of terms, and you want to find one in the middle? You could write out the sequence longhand, but there's a much easier way. You use the arithmetic sequence formula.

How to Derive the Arithmetic Sequence Formula

If you denote the first term in an arithmetic sequence by the letter ​a​, and you let the common difference between terms be ​d​, you can write the sequence in this form:

a, (a + d), (a + 2d), (a +3d), . . .

If you denote the nth term in the sequence as ​xn​, you can write a general formula for it:

x_n = a + d(n - 1)

Use this to find the 10th term in the sequence 3, 6, 9, 12, . . .

x_{10} = 3 + 3(10 - 1) = 30

Check by writing the terms out in sequence, and you'll see that it works.

A Sample Arithmetic Sequence Problem

In many problems, you are presented with a sequence of numbers, and you have to use the arithmetic sequence formula to write a rule to derive any term in that particular sequence.

For example, write a rule for the sequence 7, 12, 17, 22, 27, . . . The common difference (​d​) is 5 and the first term (​a​) is 7. The ​n​th term is given by the arithmetic sequence formula, so all you have to do is plug in the numbers and simplify:

\begin{aligned} x_n &= a + d(n - 1) \\ &= 7 + 5(n - 1) \\ &= 7 + 5n - 5 \\ &= 2 + 5n \end{aligned}

This is an arithmetic sequence with two variables, ​xn​ and ​n​. If you know one, you can find the other. For example, if you're looking for the 100th term (​x100), then ​n​ = 100 and the term is 502. On the other hand, if you want to know which term the number 377 is, rearrange the arithmetic sequence formula solve for ​n​:

\begin{aligned} n &= \frac{x_n - 2}{5} \\ \,\\ &= \frac{377 - 2}{5} \\ \,\\ &= 75 \end{aligned}

The number 377 is the 75th term in the sequence.

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