Quick answer
An arithmetic sequence is an ordered list where each term after the first is found by adding the same constant d. The pattern is linear and predictable once you name a₁ and d.
Formula
- aₙ = a₁ + (n - 1)d
- d = a₂ - a₁
Introduction
Arithmetic sequences sit at the center of early algebra because they combine simple rules with flexible applications. You see them in savings plans, seating charts, and any story where something changes by the same amount each step.
Before you jump to calculation, it helps to separate vocabulary: a sequence is the list itself, while a sum of those terms is handled in our guide on the difference between sequences and series.
When you already know a₁, d, and how many terms to display, the calculator on the home page lists every value with subscript labels in seconds.
Definition and meaning
Formally, an arithmetic sequence is a list a₁, a₂, a₃, … such that the difference between consecutive terms is constant. That constant is the common difference d.
The word sequence emphasizes order: position matters. The fifth number is not interchangeable with the second number unless the problem explicitly allows reordering.
Students sometimes confuse "sequence" with "series." A sequence uses commas between terms; a series uses addition. Keeping that distinction clear prevents half-right answers on tests.
Meaning in context: when a word problem describes equal weekly deposits or steady raises, you are often being asked to model an arithmetic sequence even before the formula appears.
Common difference and the nth term
- d = a₂ - a₁
- aₙ = a₁ + (n - 1)d
The subtraction rule d = a₂ - a₁ is the fastest check when you only have a short list. If every neighbor pair gives the same difference, the list is arithmetic.
For any position n, the nth term formula adds (n - 1) copies of d to the start. Our arithmetic sequence formula article walks through each variable with extra examples.
The factor (n - 1) counts gaps between the first term and the nth term, not the number of terms itself. That detail is a common source of off-by-one mistakes.
Step-by-step guide
- Read the situation. Decide whether the story describes equal steps between values. If steps vary, the list may not be arithmetic.
- Identify a₁. Locate the first term in the order given. In a table, confirm whether the first row is truly the starting value.
- Find d. Subtract consecutive terms. For deeper pattern checks, see our guide on finding common difference.
- Label positions. Use subscripts a₁, a₂, a₃ so you do not confuse position with total count.
- Verify on paper or with the tool. List a few terms manually, then compare with the calculator if you need a longer list.
Real-life example
Suppose you deposit $50 into a simple tracking sheet every week starting from $200 in the account. Ignoring interest, the balance after each deposit forms 200, 250, 300, 350, …
Here a₁ = 200 and d = 50. The list grows linearly because the change between weeks is constant.
If the problem later asks for a total deposited, you switch from sequence thinking to series thinking. The list still helps you identify a₁ and the last term you need.