About the Number Sequence Calculator
A sequence is a list of numbers that follows a rule. In an arithmetic sequence you add the same amount each time, such as 2, 5, 8, 11. In a geometric sequence you multiply by the same amount each time, such as 3, 6, 12, 24. This calculator takes the first term, the common difference or ratio and the number of terms, and gives the nth term, the sum of the first n terms, the formula, and a list of the first twenty terms. For a geometric sequence with a ratio between -1 and 1 it also shows the sum if the sequence went on forever.
Updated October 2026
How to use the Number Sequence Calculator
- 1Choose arithmetic or geometric.
- 2Enter the first term of the sequence.
- 3Enter the common difference (arithmetic) or common ratio (geometric).
- 4Enter how many terms you want to count, up to 10,000.
- 5Read the nth term, the total, the formula and the list of first terms.
Spotting the type of sequence
Look at how the numbers grow. If the gap between neighbours is the same, the sequence is arithmetic: 10, 14, 18, 22 grows by 4 each step. If each term is the previous one multiplied by the same factor, it is geometric: 100, 50, 25 halves each step. If neither holds, it may be quadratic, Fibonacci or something else entirely, and the formulas here will not apply.
Everyday uses
- Savings plans that add a fixed amount every month are arithmetic.
- Compound growth, depreciation and bouncing balls are geometric.
- Seating rows that gain a fixed number of seats each row are arithmetic.
- Population growth at a fixed percentage is geometric, with ratio 1 plus the rate.
Working out sums quickly
A famous story says the young Gauss added the whole numbers from 1 to 100 by pairing them: 1 plus 100, 2 plus 99, and so on, giving 50 pairs of 101, which is 5,050. The arithmetic sum formula is that same idea. For compound growth over several periods, use the Compound Interest Calculator and the CAGR Calculator. The Fibonacci Calculator covers the famous recursive sequence, and the Factorial Calculator and Exponent Calculator help with the powers that appear in geometric terms.
Features
- Arithmetic and geometric sequences
- The nth term and the formula in words
- Sum of the first n terms
- Sum to infinity when a geometric series settles
- List of the first twenty terms
- Works with decimals and negative numbers
- Runs in your browser
Frequently asked questions
What is the formula for an arithmetic sequence?
The nth term is the first term plus (n minus 1) times the common difference. For the sequence starting at 2 and adding 3 each time, the 10th term is 2 plus 9 times 3, which is 29. The sum of the first n terms is n times the first plus last term, divided by 2, so the first ten terms add up to 155.
What is the formula for a geometric sequence?
The nth term is the first term times the ratio raised to the power (n minus 1). Starting at 3 and doubling, the 5th term is 3 times 2 to the power 4, which is 48. The sum of the first n terms is the first term times (1 minus ratio to the power n), divided by (1 minus the ratio).
How do I find the common difference or ratio?
For an arithmetic sequence, subtract any term from the one after it. For a geometric sequence, divide any term by the one before it. If you have the sequence 5, 8, 11, the difference is 3. If you have 2, 6, 18, the ratio is 3. Check at least two pairs to be sure the pattern is steady.
When does a geometric series have a sum to infinity?
Only when the common ratio is between -1 and 1, so the terms shrink toward zero. In that case the sum to infinity is the first term divided by (1 minus the ratio). The sequence 1, 0.5, 0.25, and so on adds up to 2. If the ratio is 1 or larger in size, the total just grows without limit.
Can the difference or ratio be negative or a fraction?
Yes. A negative difference counts down, such as 20, 17, 14. A negative ratio makes the terms alternate in sign, such as 1, -2, 4, -8. A fractional ratio such as 0.5 makes the terms get smaller. All are supported, and a ratio of exactly zero is rejected because it is not a useful sequence.
How is this different from a Fibonacci sequence?
In a Fibonacci sequence each term is the sum of the two before it, which is neither arithmetic nor geometric. This calculator handles the two steady-rule types. For Fibonacci numbers use the Fibonacci Calculator.