🔢 Arithmetic Sequence Calculator

What is the Arithmetic Sequence Calculator?

An arithmetic sequence (also called an arithmetic progression) is built on one simple idea: you start with a number and keep adding the same value to get the next term. That fixed step is the common difference, written d. If the first term is a, the sequence runs a, a + d, a + 2d, a + 3d, and so on. When d is positive the sequence grows; when d is negative it shrinks; when d is zero every term is the same. The position of a term is its index, usually written n, starting at 1.

Two formulas do almost all the work. The nth term is a + (n - 1)d, because to reach the nth term you start at a and take n - 1 steps of size d. The sum of the first n terms, written Sn, is n/2 x (2a + (n - 1)d). A neat way to see the sum formula is that the average of the first and last term, (a + L)/2 where L is the last term, multiplied by the number of terms n, gives the total. This is the same trick the young Carl Gauss is said to have used to add 1 to 100 in seconds.

Arithmetic sequences differ from geometric sequences, where you multiply by a fixed ratio instead of adding a fixed difference. A geometric sequence like 2, 6, 18, 54 grows by a factor of 3 each step, while an arithmetic one like 2, 6, 10, 14 grows by adding 4 each step. Knowing which pattern you have tells you which formula to use, so always check whether consecutive terms differ by a constant (arithmetic) or share a constant ratio (geometric) before you calculate.

When to use it

• Finding a far-off term, such as the 50th term, without writing out the whole list by hand.
• Adding up a long run of evenly spaced numbers, like total seats across rows that grow by a fixed amount.
• Checking homework answers in algebra or pre-calculus where sequences and series appear.
• Modelling simple linear patterns, such as fixed monthly savings or a steady production increase per week.

How to use the Arithmetic Sequence Calculator

1. Enter the first term (a), the value the sequence starts with.
2. Enter the common difference (d), the fixed amount added between consecutive terms (use a negative number for a decreasing sequence).
3. Enter the number of terms (n) you want, as a whole number of 1 or more.
4. Read off the nth term, the sum of those n terms, and the listed sequence below.

Formula & method

Worked examples

nth term and running sum for a = 5, d = 2 (sequence 5, 7, 9, 11, ...)

Common mistakes to avoid

• Using n instead of (n - 1) in the nth term. The nth term is a + (n - 1)d, not a + n x d. The first term takes zero steps from the start, so to reach term n you add d only n - 1 times. Forgetting the minus one shifts every answer by one full step of d.
• Mixing up arithmetic and geometric sequences. Arithmetic sequences add a constant difference; geometric sequences multiply by a constant ratio. If consecutive terms share a ratio rather than a difference, these formulas do not apply, you need the geometric ones instead.
• Forgetting that d can be negative or zero. A decreasing sequence has a negative common difference, for example d = -7 gives 100, 93, 86, and so on. Entering it as positive flips the direction and gives the wrong terms and sum.
• Counting the number of terms incorrectly. When a sequence is given by its first and last terms, the count is n = (L - a)/d + 1, not just (L - a)/d. The plus one includes the starting term itself, and leaving it off undercounts by one.

Glossary

Arithmetic sequence

A list of numbers in which each term differs from the previous one by a fixed amount, the common difference.

Common difference (d)

The constant value added to one term to get the next. It can be positive, negative, or zero.

First term (a)

The number the sequence starts with, the term at position n = 1.

nth term

The term at position n in the sequence, given by a + (n - 1)d.

Series (sum)

The result of adding the terms of a sequence together. The sum of the first n terms is written Sn.

Arithmetic progression

Another name for an arithmetic sequence, common in many textbooks.

Frequently asked questions