Introduction
An arithmetic sequence is a list of numbers where you add the same value each time to get the next number. For example, 2, 5, 8, 11 is an arithmetic sequence because you add 3 each time. That repeated value is called the common difference. This Arithmetic Sequence Calculator helps you find any term in the sequence, the common difference, or the sum of a set number of terms. Just enter the values you know, and the calculator does the rest. It is a quick and easy way to solve arithmetic sequence problems without doing the math by hand.
How to use our Arithmetic Sequence Calculator
Enter the details of your arithmetic sequence below, and this calculator will find the nth term, the sum of the sequence, and the common difference for you.
First Term (a₁): Type the first number in your arithmetic sequence. This is the starting value of your pattern.
Common Difference (d): Enter the amount that is added to each term to get the next term. This number stays the same throughout the sequence. It can be positive, negative, or zero.
Number of Terms (n): Enter which term in the sequence you want to find. For example, if you want the 10th term, type 10.
What Is an Arithmetic Sequence?
An arithmetic sequence is a list of numbers where the difference between each number and the next is always the same. This difference is called the common difference. For example, in the sequence 2, 5, 8, 11, 14, the common difference is 3 because you add 3 each time to get the next number.
How Arithmetic Sequences Work
Every arithmetic sequence has two key parts: a first term (the starting number) and a common difference (the amount you add or subtract each step). If the common difference is positive, the sequence grows larger. If it's negative, the sequence gets smaller. For example, 20, 17, 14, 11 has a common difference of −3.
The Arithmetic Sequence Formula
You can find any term in an arithmetic sequence using this formula:
aₙ = a₁ + (n − 1) × d
Here, aₙ is the term you want to find, a₁ is the first term, n is the position of the term, and d is the common difference. For example, if your first term is 3 and your common difference is 5, the 10th term would be 3 + (10 − 1) × 5 = 48.
Sum of an Arithmetic Sequence
You can also add up all the terms in an arithmetic sequence without adding them one by one. The formula for the sum of the first n terms is:
Sₙ = n/2 × (a₁ + aₙ)
This works because the average of the first and last term, multiplied by the number of terms, gives you the total sum. This shortcut was famously used by the mathematician Carl Friedrich Gauss when he was just a child.
Where Are Arithmetic Sequences Used?
Arithmetic sequences show up in everyday life more than you might think. They appear in things like saving a fixed amount of money each week, counting seats in a stadium that increase by the same number per row, or calculating evenly spaced time intervals. They are also a building block for more advanced math topics like arithmetic series and linear functions.
Arithmetic sequences are closely related to other mathematical concepts you may encounter. The Fibonacci sequence is a well-known pattern that, unlike arithmetic sequences, adds the two previous terms together rather than a fixed value. For sequences that grow by multiplication rather than addition, you would be working with geometric sequences, which tie into concepts like exponents and logarithms.