Arithmetic Sequence Calculator

Arithmetic sequences show up in places you might not even think about — adding a set amount to your savings each month, planning out a payment schedule, or watching how a pattern of numbers grows step by step. These sequences are built on a simple idea: each number in the series is spaced apart by the same fixed amount, called the common difference.

What Is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers that follows a clear pattern: each new number is created by adding or subtracting the same amount every time. That “same amount” is called the common difference.

Take this example: 5, 10, 15, 20, 25…

• The first term (a₁) is 5.

• The common difference (d) is 5, since every number increases by 5.

These patterns are far from just textbook exercises. They can help with real-world questions — like figuring out how much money you’ll have after a year of saving the same amount each week, or how many steps it takes to reach a goal if you increase your effort bit by bit.

Because the numbers move in a steady line, arithmetic sequences are straightforward to predict and work with. That’s why they’re a go-to concept in math lessons, and why they pop up so often in practical problem-solving.

The Formula Behind Arithmetic Sequences

Arithmetic sequences might look simple, but there’s a straightforward set of formulas that make working with them even easier. These formulas help you quickly find any term in the sequence or add up a whole chunk of terms without writing them all out.

Finding Any Term in the Sequence

The formula to find the nth term (any position in the sequence) is:

aₙ = a₁ + (n − 1)d

Here’s what each part means:

• aₙ – the term you’re looking for (for example, the 10th number).

• a₁ – the first term in the sequence.

• d – the common difference (how much each term changes).

• n – the position of the term you’re finding.

Adding Up Multiple Terms

If you need to find the sum of the first n terms, there’s another handy formula:

Sₙ = (n ÷ 2) × [2a₁ + (n − 1)d]

This formula lets you add up the numbers without listing every single one.

A Quick Example

Let’s say you have the sequence: 3, 6, 9, 12, …, and you want to find:

1. The 10th term.

2. The sum of the first 10 terms.

First, identify what you know:

• a₁ = 3 (first term)

• d = 3 (each term increases by 3)

• n = 10 (we want the 10th term and the sum of 10 terms)

Step 1: Find the 10th term

a₁₀ = 3 + (10 − 1)(3) = 3 + 27 = 30

So, the 10th term is 30.

Step 2: Find the sum of the first 10 terms

S₁₀ = (10 ÷ 2) × [2(3) + (10 − 1)(3)]

S₁₀ = 5 × [6 + 27] = 5 × 33 = 165

So, the sum of the first 10 terms is 165.

Arithmetic vs. Other Sequences

Not every sequence grows in the same way. Arithmetic sequences move at a steady pace, adding or subtracting the same amount each time. But there are other types of sequences that work very differently.

• Geometric Sequences

Instead of adding a fixed amount, a Geometric Sequence changes by multiplying by the same factor each step. For example, 2, 4, 8, 16, 32… doubles each time instead of adding a set number. These are common when modeling things like population growth, investments with interest, or anything that grows (or shrinks) exponentially.

• Fibonacci Sequences

The Fibonacci Sequence doesn’t use addition or multiplication in a straightforward way. Instead, each term is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8…. You’ll spot this pattern in everything from computer algorithms to the spirals of seashells.

If you’re not sure which type fits your problem, you can explore our other tools:

• Geometric Sequence Calculator

• Fibonacci Sequence Calculator

• Or check broader tools like our Number Sequence Calculator and Math Calculator.