🔢 Arithmetic & Geometric Sequence Calculator
Enter the first term, the common difference or ratio, and how many terms to get the nth term, the sum and the sequence itself.
nth term (aₙ)
29
Sum of first n (Sₙ)
155
First terms
2, 5, 8, 11, 14, 17, 20, 23, 26, 29
An arithmetic sequence adds a fixed common difference d each step: aₙ = a₁ + (n−1)d, and the sum of the first n terms is Sₙ = n⁄2 · (2a₁ + (n−1)d). 🔒 In your browser.
How the arithmetic & geometric sequence calculator works
An arithmetic sequence changes by a fixed amount each step, the common difference d, so the nth term is aₙ = a₁ + (n − 1)d and the sum of the first n terms is Sₙ = n⁄2 · (2a₁ + (n − 1)d). A geometric sequence multiplies by a fixed common ratio r each step, so aₙ = a₁ · r^(n − 1) and the partial sum is Sₙ = a₁(1 − rⁿ)⁄(1 − r) (and simply a₁ · n when r = 1). Pick the type, enter the three values, and the tool returns the nth term, the running sum and the first terms.
Use the arithmetic mode for anything that grows by a constant amount (savings of a fixed monthly deposit, rows of seats increasing by a set number) and geometric for constant-ratio growth or decay (compound interest, halving, population doubling). For a geometric series with |r| < 1, the partial sums approach a finite limit a₁⁄(1 − r) as n grows.
Frequently asked questions
How do I find the nth term of a sequence?
For an arithmetic sequence, aₙ = a₁ + (n − 1)d, where d is the common difference. For a geometric sequence, aₙ = a₁ · r^(n − 1), where r is the common ratio. Enter your values and the tool computes it exactly.
What is the difference between arithmetic and geometric?
An arithmetic sequence adds the same number each step (2, 5, 8, 11…, difference 3); a geometric sequence multiplies by the same number each step (3, 6, 12, 24…, ratio 2). One grows linearly, the other exponentially.
How do I calculate the sum of a sequence?
Arithmetic: Sₙ = n⁄2 · (2a₁ + (n − 1)d), or n × the average of the first and last terms. Geometric: Sₙ = a₁(1 − rⁿ)⁄(1 − r). This tool shows the sum of the first n terms automatically.
What is the common difference or common ratio?
The common difference d is what you add each step in an arithmetic sequence (found by subtracting any term from the next). The common ratio r is what you multiply by each step in a geometric sequence (found by dividing any term by the previous one).
Does a geometric series have a finite sum?
The infinite sum converges only when |r| < 1, to a₁⁄(1 − r). If |r| ≥ 1 the terms don't shrink and the sum grows without bound. This calculator gives the sum of a specified finite number of terms either way.