An arithmetic sequence is a list of numbers with a constant gap between consecutive terms, called the common difference. This lesson covers the definition, the explicit and recursive formulas for finding any term, and worked examples that build the skills needed for arithmetic series and progression problems.
What Is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers arranged so that the difference between any two consecutive terms is always the same. That fixed gap is called the common difference, usually written as \(d\). For example, in the sequence \(3, 7, 11, 15, 19, \dots\), each term is \(4\) more than the one before it, so \(d = 4\).
To check whether a sequence is arithmetic, subtract each term from the term that follows it. If you always get the same number, the sequence is arithmetic. If the difference changes, or if terms are being multiplied instead of added, the sequence is not arithmetic; you may instead be looking at a geometric sequence, where each term is found by multiplying by a fixed ratio.
The Common Difference
The common difference can be positive, negative, or even a fraction:
If \(d > 0\), the sequence increases, like \(2, 5, 8, 11, \dots\) where \(d = 3\).
If \(d < 0\), the sequence decreases, like \(20, 15, 10, 5, \dots\) where \(d = -5\).
If \(d = 0\), every term is identical, like \(6, 6, 6, 6, \dots\).
To find \(d\) from any two consecutive terms \(a_n\) and \(a_{n+1}\), use \(d = a_{n+1} - a_n\).
The Nth Term (Explicit) Formula
The most useful tool for working with arithmetic sequences is the explicit formula, which lets you calculate any term directly without listing every value before it:
\(a_n = a_1 + (n-1)d\)
Here, \(a_n\) is the term you want to find, \(a_1\) is the first term, \(n\) is the position of the term in the sequence, and \(d\) is the common difference.
Plotting term value against term number shows why arithmetic sequences form a straight line: each step up in \(n\) adds the same amount, \(d\), to the value.
Worked Example 1: Finding a Specific Term
Find the 12th term of the sequence \(5, 9, 13, 17, \dots\)
Here \(a_1 = 5\) and \(d = 4\). Substitute \(n = 12\):
\(a_{12} = 5 + (12 - 1)(4) = 5 + 44 = 49\)
So the 12th term is \(49\).
Worked Example 2: Finding the Common Difference and First Term
The 4th term of an arithmetic sequence is \(18\) and the 9th term is \(38\). Find \(d\) and \(a_1\).
Since both terms use the same formula, subtract to eliminate \(a_1\):
\(a_9 - a_4 = (a_1 + 8d) - (a_1 + 3d) = 5d\)
\(38 - 18 = 5d \implies d = 4\)
Now substitute back into \(a_4 = a_1 + 3d\): \(18 = a_1 + 3(4)\), so \(a_1 = 6\).
The Recursive Formula
Instead of jumping straight to any term, a recursive formula defines each term using the one right before it:
\(a_n = a_{n-1} + d, \quad \)with a given \( a_1\)
For the sequence \(5, 9, 13, 17, \dots\), the recursive rule is \(a_n = a_{n-1} + 4\) with \(a_1 = 5\). This is handy for describing the pattern, but the explicit formula is faster when you need a term far down the list, like the 200th term, since it avoids calculating every term in between.
Arithmetic Sequences vs. Geometric Sequences
It is easy to confuse arithmetic and geometric patterns, so it helps to check the operation connecting the terms. Arithmetic sequences are built by repeated addition of a common difference, while geometric sequences are built by repeated multiplication by a common ratio. Comparing the two types side by side, including how their averages behave, is covered in more detail on the arithmetic mean vs geometric mean lesson.
Where This Leads: Arithmetic Series
Once you can identify an arithmetic sequence and find its terms, a natural next question is how to add a large group of those terms together efficiently, for example, the sum of the first 50 terms. That process, including the sum formula and its derivation, is covered fully on the arithmetic series lesson, which builds directly on the skills in this topic.
Quick Recap
An arithmetic sequence increases or decreases by the same amount, \(d\), between consecutive terms. Use \(a_n = a_1 + (n-1)d\) to find any term directly, or the recursive rule \(a_n = a_{n-1} + d\) to describe the pattern step by step. Practicing with both positive and negative common differences will help these formulas feel automatic before moving on to arithmetic series.