A sequence is arithmetic if the difference between every pair of consecutive terms is exactly the same. That constant gap, called the common difference, is the single defining feature. Whether the sequence goes up, goes down, or stays flat, the rule is the same: subtract any term from the one that follows it, and you always get the same number.
The Common Difference
The common difference is usually represented by the letter d. You find it by subtracting any term from the next one: d = a(n+1) − a(n). In the sequence 3, 5, 7, 9, the common difference is 2 because 5 − 3 = 2, 7 − 5 = 2, and 9 − 7 = 2. Every consecutive pair produces the same result.
The common difference can be positive, negative, or zero. A positive difference means the sequence increases (2, 5, 8, 11). A negative difference means it decreases (20, 15, 10, 5, with d = −5). A common difference of zero gives you a flat sequence where every term is identical (4, 4, 4, 4), which technically still qualifies as arithmetic.
How to Test Any Sequence
To check whether a sequence is arithmetic, subtract each term from the one after it and see if the result stays constant. Take the sequence 10, 16, 22, 28. The differences are 6, 6, and 6, so it’s arithmetic with d = 6. Now try 2, 4, 8, 16. The differences are 2, 4, and 8, which aren’t constant, so this one is not arithmetic (it’s actually geometric, but more on that below).
This subtraction test is the only thing you need. If even one pair of consecutive terms produces a different gap, the sequence fails the test. There’s no partial credit here: the difference is either constant throughout or it isn’t.
Two Ways to Write the Rule
The Explicit Formula
The explicit formula lets you jump straight to any term without calculating all the ones before it. The standard form is a(n) = A + B(n − 1), where A is the first term, B is the common difference, and n is the position of the term you want. For the sequence 3, 5, 7, 9, the formula is a(n) = 3 + 2(n − 1). Plug in n = 50, and you get the 50th term directly: 3 + 2(49) = 101.
The Recursive Formula
A recursive formula builds the sequence one step at a time. It has two parts: the value of the first term, and a rule that says “take the previous term and add d.” Think of it like climbing a ladder. To reach the third rung, you have to step on the second rung first. Formally, it looks like this: a(1) = first term, and a(n) = a(n−1) + d. For the sequence 3, 5, 7, 9, the recursive rule is a(1) = 3 and a(n) = a(n−1) + 2.
The recursive form is intuitive because it mirrors how you’d actually build the sequence by hand. The explicit form is more practical when you need the 200th term without grinding through the first 199.
Why It Graphs as a Straight Line
If you plot an arithmetic sequence on a graph, with the term number on the horizontal axis and the term’s value on the vertical axis, the points fall on a perfectly straight line. This happens because an arithmetic sequence is really a linear function applied to the counting numbers. The common difference acts as the slope of that line. A sequence with d = 3 produces a line with a slope of 3; a sequence with d = −50 produces a line sloping downward at −50.
This connection works both ways. If you graph a sequence and the points form a straight line, the sequence is arithmetic. If the points curve, it isn’t. The constant rate of change that defines a linear function is the same constant difference that defines an arithmetic sequence.
Arithmetic vs. Geometric Sequences
The most common source of confusion is mixing up arithmetic and geometric sequences. The distinction is simple: arithmetic sequences are built by adding the same number each time, while geometric sequences are built by multiplying by the same number each time. An arithmetic sequence has a common difference (d); a geometric sequence has a common ratio (r).
- Arithmetic: 5, 10, 15, 20 (adding 5 each time, d = 5)
- Geometric: 5, 10, 20, 40 (multiplying by 2 each time, r = 2)
Both sequences start at 5, but they grow in fundamentally different ways. Arithmetic sequences grow (or shrink) steadily. Geometric sequences accelerate because each multiplication compounds on the previous result. On a graph, arithmetic sequences trace a straight line while geometric sequences trace a curve.
Everyday Examples
Arithmetic sequences show up more often than you’d expect. A pool being filled by a garden hose at a constant rate creates one: if the water level rises 3 inches every hour, the depth at each hour forms an arithmetic sequence with d = 3. A stadium where each row has 4 fewer seats than the row behind it follows an arithmetic pattern with d = −4. Even stacking identical objects, like cups or chairs, produces one, because each added object increases the total height by the same amount.
Divers encounter arithmetic sequences, too. Safe descent guidelines recommend going no faster than 66 feet per minute, meaning depth at each minute follows an arithmetic progression: 0, −66, −132, −198, and so on. Any situation where something increases or decreases at a steady, unchanging rate is an arithmetic sequence in disguise.