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Arithmetic and Geometric Sequence and Series Calculator

This free calculator finds the nth term and the sum of the first n terms for arithmetic and geometric sequences. It also lists the first terms so you can see the pattern.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

A sequence is a list of numbers that follows a rule, and a series is the sum of that list. Arithmetic sequences grow by equal steps, like saving the same amount each month. Geometric sequences grow by equal factors, like compound interest.

Give the first term, the step and the number of terms, and get the last term, the sum and a preview of the list.

How to Use This Tool

  1. Choose arithmetic or geometric.
  2. Enter the first term, the common difference or ratio, and the number of terms.
  3. Press Calculate.
  4. Read the nth term, the sum and the listed terms.

How the Calculation Works

Arithmetic: the nth term is a1 + (n − 1)d and the sum is n ÷ 2 × (a1 + nth term). Geometric: the nth term is a1 × r^(n−1) and the sum is a1 × (rⁿ − 1) ÷ (r − 1), or n × a1 if r = 1.

Worked Example

Arithmetic with a1 = 3, d = 4 and n = 10: the 10th term is 39 and the sum is 210. Geometric with a1 = 2, r = 3 and n = 6: the 6th term is 486 and the sum is 728.

Common Uses

  • Savings plans and loan schedules.
  • Compound growth and decay.
  • Exam questions on progressions.
  • Pattern puzzles.

Tips and Common Mistakes

  • n must be a whole number from 1 upwards.
  • A negative d gives a decreasing sequence.
  • A ratio above 1 grows very fast, so large n can overflow.

Next, you may need the Fibonacci calculator, exponent and root calculator, logarithm calculator, percentage calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

A sequence where each term differs from the last by the same amount, such as 3, 7, 11, 15. The common difference here is 4.
A sequence where each term is the last one times a fixed number, such as 2, 6, 18, 54. The common ratio here is 3.
Arithmetic: S = n ÷ 2 × (2a1 + (n − 1)d). Geometric: S = a1 × (rⁿ − 1) ÷ (r − 1) when r is not 1.
If the ratio r is between −1 and 1, the infinite sum is a1 ÷ (1 − r). The tool shows it when it applies.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.