Arithmetic Sequences: Formulas, Rules, and Examples Explained

You use arithmetic sequences anytime you count by a set step—like adding 3 each time or skipping every other number. An arithmetic sequence is just a list of numbers that changes by the same amount each time, so you can actually predict any term or the sum of several terms with some pretty straightforward formulas.

A desk with an open notebook showing numbers in an arithmetic sequence, a hand writing with a pen, a calculator, and a laptop displaying a linear graph.

Here, I’ll walk you through the main idea, the key formulas you’ll need, and some practical ways to use them in puzzles, budgeting, or schoolwork. Stick around for short examples and some easy steps—these let you find any term or total without a ton of guesswork.

Core Concepts of Arithmetic Sequences

A person studying arithmetic sequences with a notebook, pen, calculator, and laptop on a desk.

Let’s talk about how arithmetic sequences work, how to spot the constant step between terms, how these sequences connect to straight-line patterns, and some clear examples you can check by hand.

What Is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where each term after the first comes from adding the same fixed number. That fixed number is the common difference.

If you start with a first term a and a common difference d, the terms look like: a, a + d, a + 2d, a + 3d, and so on.

You can write any term using this formula: the nth term is a + (n – 1)d.

This lets you find a far-off term without having to list out every number.

The sequence can increase, decrease, or just stay the same. If d > 0, the sequence grows. If d < 0, it shrinks. If d = 0, well, every term is just the first term again.

Understanding the Common Difference

The common difference is just the difference between each pair of consecutive terms. You find it by subtracting one term from the next: d = a2 – a1 = a3 – a2.

This quick check helps you confirm if a list is arithmetic.

The common difference stays the same throughout the sequence. If you know the first term and a later term, you can find d with d = (an – a1) / (n – 1).

d can be any real number—positive, negative, whole number, or even a fraction.

Write d clearly when you describe a sequence, since it controls the step size and direction.

Identifying Linear Sequences

Arithmetic sequences line up perfectly with linear functions, where the input is the term number n. Think of n as x and the term as f(n) = d·n + b.

If you graph term number versus value, you get a straight line.

To check if a sequence is linear, just see if the differences between terms are always the same.

If they are, you’ve got a linear sequence, and you can use slope-like reasoning. The common difference works just like the slope.

Linear sequences make it easier to use algebra—like solving for n, finding the first term, or predicting future terms.

Recognizing Examples of Arithmetic Sequences

Look for a constant step when you see a list. Some classic examples:

  • 3, 6, 9, 12, … (first term a = 3, d = 3)
  • 80, 75, 70, 65, … (a = 80, d = -5)
  • 5, 5, 5, 5, … (a = 5, d = 0)

Test any sequence by subtracting one term from the next. If the result always matches, you’ve got an arithmetic sequence.

Use the nth-term formula a + (n – 1)d to generate more terms or find a term far out in the list.

Here’s a quick checklist:

  • Find the first term a.
  • Work out d from two consecutive terms.
  • Check that d stays the same across several steps.
  • Use the formula to make or check terms.

Arithmetic Sequence Formulas and Applications

Let’s look at how to write rules for any term, use the previous term to get the next, and add up lots of terms. You’ll see formulas for the nth term, for the sum of n terms, and some clear steps to tackle common problems.

Explicit Rule for the nth Term

Use the explicit rule if you want a direct formula for any term by its place in the sequence.

If the first term is a and the common difference is d, the nth term is:

  • an = a + (n − 1)d

Plug in your numbers for a, d, and n to get the value of the nth term.

Example: a = 3, d = 5, n = 6 → a6 = 3 + 5(6 − 1) = 28.

You can rearrange the formula if you need to solve for n or d and you know other values.

  • To find d: d = (an − a)/(n − 1).
  • To find n when you know an: n = (an − a)/d + 1.

Keep your numbers and units straight. This rule works whether d is positive, negative, or zero.

Recursive Rule and Recursive Formula

The recursive rule uses the previous term to get the next one. People also call it the term-to-term rule.

It looks like this:

  • a1 = a (given)
  • an = an−1 + d (for n ≥ 2)

This tells you how each term changes from the one before.

Use the recursive rule when you know a starting value and want to build the list step by step.

Example: a1 = 10, d = −2 → a2 = 8, a3 = 6, a4 = 4.

Recursive formulas help in programming or when you build a sequence one step at a time.

They don’t give you a direct value for a distant term unless you work out all the previous ones or switch to the explicit rule.

Finding the Sum: Arithmetic Series

The sum of the first n terms (an arithmetic series) gives you the total when you add terms from term 1 to term n.

You’ve got two common formulas, depending on what you know:

  • If you know a and d: Sn = n/2 [2a + (n − 1)d]
  • If you know a and an: Sn = n/2 [a + an]

Both formulas get you to the same place, so use whichever fits your information.

Example: a = 4, d = 3, n = 5 → an = 4 + 3(4) = 16, Sn = 5/2 [4 + 16] = 50.

You can find Sn by pairing terms (first plus last) and multiplying by n/2.

These formulas help with finance problems, counting totals, or any situation where you keep adding a constant step.

Practical Applications and Problem Solving

You can use these formulas for things like salary increases, seating rows, or even steps in a staircase. For salary, just set a as your starting pay, d as the yearly raise, n as the number of years, and Sn will give you the total pay over those years.

Here’s a quick example. Let’s say a is $30,000, d is $2,000, and n is 4. Plug those in: 4/2 [2(30000) + 3(2000)]—that comes out to $126,000.

When you tackle contest problems, figure out a, d, and what you actually need—usually an or Sn. I like to make a table with the first term, d, which formula I used, and the result. It really helps keep everything straight and, honestly, saves you from silly mistakes.

If they give you two nonconsecutive terms, just use d = (an − am)/(n − m) to find d first. After that, you can get a or any other term you need.

Use the explicit rule when you want a direct answer. If you’re modeling something step by step, the recursive rule feels more natural.

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