Arithmetic Sequence Calculator

What this calculator does

An arithmetic sequence is a list of numbers that goes up, or down, by the same step every time: 5, 8, 11, 14, and so on, always adding 3. This tool works with one of those from three simple facts about it.

Give it the first term, the step between terms, and how far along you want to go, and it lists the sequence, tells you that final term, and adds them all up for you. It runs right here in the browser.

Using the calculator

1. Enter the first term (a1), where the sequence begins.
2. Enter the common difference (d), the step added each time. A negative d makes the sequence count down.
3. Enter the term number (n), how many terms you want.
4. Pick a precision if your numbers have decimals, then press Calculate.

You get the sequence, the nth term, and the sum of the first n terms, all together.

What an arithmetic sequence is

The defining feature is a constant difference. You get from one term to the next by adding the same amount every single time, and that amount is called the common difference, d. In 5, 8, 11, 14 the common difference is 3. If it were -2, the sequence would fall: 20, 18, 16, 14.

These turn up all over the place: counting by fives, the seats in each row of a theatre that grows by a fixed number, a savings plan where you put away the same amount each month. Anything that changes by a steady step is arithmetic.

Finding any term: the nth term formula

You could reach the 100th term by adding d ninety-nine times, but there is a shortcut:

a_n = a1 + (n − 1) × d

where a1 is the first term, d is the common difference, and n is the position you want. The reason it is (n − 1) and not n is that the first term already sits there for free; to reach term number n you only add the step d the remaining (n − 1) times. So the 10th term of 5, 8, 11 is 5 + 9 × 3 = 32, no counting required.

Adding them up: the sum formula

To add the first n terms, there is a lovely shortcut that goes back to a trick Gauss is said to have used as a schoolboy. Pair the first term with the last, the second with the second-to-last, and so on. Every pair adds up to the same total, so the whole sum is just that pair-total times the number of pairs:

S_n = (n ÷ 2) × (first term + last term)

which, writing the last term out in full, is the form the tool uses: S_n = (n ÷ 2) × (2a1 + (n − 1)d). This is exactly why the sum of the whole numbers from 1 to n comes to n(n + 1) ÷ 2: that is just this formula for the simplest arithmetic sequence of all, 1, 2, 3, 4.

A worked example

Take a first term of 5, a common difference of 3, and 6 terms.

1. The sequence: 5, 8, 11, 14, 17, 20.
2. The 6th term, by the formula: 5 + (6 − 1) × 3 = 5 + 15 = 20, which matches the last term in the list.
3. The sum: (6 ÷ 2) × (5 + 20) = 3 × 25 = 75. And indeed 5 + 8 + 11 + 14 + 17 + 20 = 75.

On decimals and precision

If your first term or common difference has decimals, ordinary browser maths can leave tiny rounding specks in the last digits of a long sequence. This tool avoids that by running the arithmetic through Decimal.js, a library built for exact decimal work, and the precision dropdown sets how many decimal places it carries. For whole-number sequences you can leave it alone; for decimals where the fine detail matters, turn it up.

Questions people ask

How do I find the nth term of an arithmetic sequence?

Use a_n = a1 + (n − 1) × d. Start from the first term and add the common difference (n − 1) times.

How do I find the sum of an arithmetic sequence?

Multiply the number of terms by the average of the first and last terms: S_n = (n ÷ 2) × (first + last).

What is the common difference?

The fixed amount added to get from one term to the next. Subtract any term from the one after it to find it.

Can the sequence go down?

Yes. A negative common difference makes each term smaller than the last, so the sequence counts downward.

How is this different from a geometric sequence?

An arithmetic sequence adds a fixed amount each step; a geometric sequence multiplies by a fixed amount. Adding versus multiplying is the whole difference.

References

A note on where this comes from. A sequence with a constant difference between terms is an arithmetic progression, one of the oldest and most standard objects in mathematics, and its nth-term and sum formulas are classical results found in any algebra reference, including Wolfram MathWorld. For further reading, see Arithmetic progression.

1. Weisstein, Eric W. "Arithmetic Progression", Wolfram MathWorld, on the nth term and the sum of an arithmetic sequence.
2. Decimal.js, an arbitrary-precision decimal library, used here so decimal sequences and sums stay exact.