Overview
An arithmetic sequence is a list of numbers where each term changes from the previous one by a fixed amount called the common difference. A familiar example is 2, 5, 8, 11, … with difference 3. Any term and the sum of the first n terms follow simple formulas.
Formula
aₙ = a₁ + (n − 1)d, and Sₙ = n(a₁ + aₙ) ÷ 2 = n(2a₁ + (n − 1)d) ÷ 2.
How it works
Identify the inputs
The first term a₁, common difference d and number of terms n.
Compute the nth term
Substitute into aₙ = a₁ + (n − 1)d.
Compute the sum
Use Sₙ = n(a₁ + aₙ) ÷ 2.
Verify
List the first terms and confirm the difference is constant.
Worked examples
Example 1: a₁ = 2, d = 3, n = 10
a₁₀ = 2 + 9×3 = 29, and the sum = 10×(2 + 29) ÷ 2 = 155.
Example 2: a₁ = 10, d = −2, n = 5
The terms are 10, 8, 6, 4, 2, with sum 30.
Frequently asked questions
What if the difference is zero?
All terms become equal, and the sum is n × a₁.
Can I use fractions?
Yes. The calculator accepts decimals and negative numbers.
What is the difference between a sequence and a series?
A sequence is a list of terms, while a series is the sum of those terms.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.