Arithmetic Sequence Calculator
Work with an arithmetic sequence — a list of numbers that increases or decreases by a fixed common difference. Enter the first term, the common difference, and how many terms you want, and the calculator returns the nth term aₙ = a₁ + (n−1)d and the sum of the first n terms Sₙ = n⁄2 × (2a₁ + (n−1)d), along with a preview of the opening terms.
Formula
aₙ = a₁ + (n−1)d · Sₙ = n/2 × (2a₁ + (n−1)d)
Example: a₁ = 3, d = 4, n = 10
Inputs
- First Term (a₁): 3
- Common Difference (d): 4
- Number of Terms (n): 10
The sequence is 3, 7, 11, 15, … The 10th term is 3 + (10−1) × 4 = 39, and the sum of the first 10 terms is 10⁄2 × (2 × 3 + 9 × 4) = 5 × 42 = 210.
Frequently asked questions
What is an arithmetic sequence?
It is a list of numbers where each term differs from the previous one by the same fixed amount, called the common difference (d). For example, 3, 7, 11, 15 has d = 4.
How do I find the nth term?
Use aₙ = a₁ + (n−1)d: start from the first term and add the common difference (n−1) times.
How is the sum of the sequence calculated?
The sum of the first n terms is Sₙ = n⁄2 × (2a₁ + (n−1)d), which is also n times the average of the first and last terms.
Can the common difference be negative?
Yes. A negative d makes the sequence decrease, for example 20, 17, 14, 11 with d = −3.
What is the difference between a sequence and a series?
The sequence is the ordered list of terms; the series (sum) is what you get by adding those terms together. This calculator gives you both.
Does n have to be a whole number?
Yes — n counts how many terms there are, so it must be a positive integer. The calculator rounds it to the nearest whole number.