Arithmetic Sequence Calculator
Find any term, sum, or missing value in an arithmetic sequence — instantly, with a full step-by-step solution, term list, and visual chart. Built for students, teachers, and anyone who needs to show their working.
What Is an Arithmetic Sequence?
An arithmetic sequence (also called an arithmetic progression) is any list of numbers where the gap between each consecutive pair is always the same. That fixed gap is called the common difference, usually written as d.
Three quick examples to make it concrete:
The common difference can be any real number — positive, negative, a fraction, or zero. That flexibility is what makes arithmetic sequences so common in real-world problems, from saving money each week to calculating distance in uniform motion.
Arithmetic Sequence Formulas Explained
There are two core formulas. Every result this calculator produces comes from one or both of these.
a₇ = 2 + (7−1) × 5 = 2 + 30 = 32
S₇ = 7/2 × (4 + 30) = 3.5 × 34 = 119
There's also an equivalent sum formula worth knowing: Sₙ = n/2 × (a₁ + aₙ). This version is useful when you already know the first and last term but not the common difference — which is exactly the kind of problem the percentage calculator approach uses: find what you can directly, then build to what you need.
How to Use This Calculator
Most arithmetic sequence tools online only let you enter a first term and common difference. This one lets you solve for any unknown — including finding the common difference from two known terms, or finding the first term when you know a later term.
Find the nth Term
Enter the first term (a₁), common difference (d), and which term position (n) you want. The calculator returns aₙ with full working.
Find the Partial Sum
Enter a₁, d, and how many terms (n) to add. Gets the cumulative total Sₙ — useful for adding up savings, scores, or any evenly-spaced series.
Find the Common Difference
Know two terms and their positions? Enter them and the calculator solves for d — great for reverse-engineering a sequence from partial data.
Find the First Term
Enter any known term, its position, and d. The calculator works backwards to find a₁ — the starting point of the whole sequence.
Worked Examples
These are the kinds of arithmetic sequence problems that appear most in school exams, competitive tests, and everyday life.
Real-World Uses of Arithmetic Sequences
Arithmetic sequences aren't just a classroom topic. They describe a surprising number of real patterns in everyday life and professional work.
Arithmetic vs Geometric Sequence — Key Differences
Students often confuse these two. The difference is simple: arithmetic sequences add a fixed number each step; geometric sequences multiply by a fixed number each step.
| Feature | Arithmetic | Geometric |
|---|---|---|
| Operation | Add constant (d) | Multiply by constant (r) |
| Example | 2, 5, 8, 11, 14 | 2, 6, 18, 54, 162 |
| nth term | a₁ + (n−1)d | a₁ × rⁿ⁻¹ |
| Graph shape | Straight line | Exponential curve |
| Sum to infinity | Always ∞ | Finite if |r| < 1 |
| Common in | Linear growth, savings | Compound interest, population |
Frequently Asked Questions
What is an arithmetic sequence?
An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is always the same constant value, called the common difference. For example: 3, 7, 11, 15, 19 — each term is 4 more than the previous one, so the common difference (d) is 4. The sequence can be finite (a set number of terms) or infinite.
What is the formula for the nth term of an arithmetic sequence?
The nth term formula is aₙ = a₁ + (n − 1) × d, where a₁ is the first term, d is the common difference, and n is the position of the term you want to find. To use it: multiply (n−1) by d, then add the result to a₁. The calculator above applies this formula automatically and shows every step.
How do you find the sum of an arithmetic sequence?
Use the partial sum formula: Sₙ = n/2 × (2a₁ + (n−1)d). An equivalent version is Sₙ = n/2 × (a₁ + aₙ), which is easier when you already know the last term. The logic behind it: pair the first and last term, the second and second-to-last, and so on — each pair sums to the same value, making calculation fast.
Can the common difference be negative?
Yes. A negative common difference simply means the sequence is decreasing — each term is smaller than the one before it. For example: 100, 90, 80, 70… has d = −10. The nth term formula and sum formula work exactly the same way whether d is positive, negative, or zero.
What is the sum of the first 100 natural numbers?
The sum of 1 + 2 + 3 + … + 100 = 5,050. Using the formula: S₁₀₀ = 100/2 × (1 + 100) = 50 × 101 = 5,050. This is famously attributed to mathematician Carl Friedrich Gauss, who reportedly solved it as a schoolchild in seconds by noticing that pairing 1 with 100, 2 with 99, etc. always gives 101.
How do I find the common difference if I know two terms?
Use: d = (aₙ − aₘ) / (n − m), where aₙ and aₘ are two known terms at positions n and m. For example, if the 3rd term is 11 and the 7th term is 27: d = (27 − 11) / (7 − 3) = 16 / 4 = 4. Use the "Common Difference" mode in the calculator above to do this automatically.
What is the difference between an arithmetic sequence and an arithmetic series?
An arithmetic sequence is the ordered list of terms: a₁, a₂, a₃, … An arithmetic series is the sum of those terms: a₁ + a₂ + a₃ + … The sequence describes the pattern; the series describes what you get when you add the terms together. This calculator handles both.