About the Arithmetic Sequence Calculator
This arithmetic sequence calculator works with any sequence that goes up or down by the same amount each step, such as 3, 7, 11, 15… Enter the first term and the common difference, choose which term you want, and it returns the nth term, the sum of the first n terms (the arithmetic series), a list of the terms and a running-total chart.
It is useful for algebra and pre-calculus homework, for checking formula answers, and for everyday problems that grow in equal steps: seats in rows of a theatre, a savings plan that adds a fixed extra amount each month, stacked objects, or salary scales with fixed annual raises. The “check a value” box tells you whether a number belongs to the sequence and, if so, its position.
The common difference can be negative (a decreasing sequence) or a decimal. If you only know two terms, subtract them and divide by the gap in positions to get the common difference, then enter it here.
With the default inputs, the nth term (aₙ) is 39. Change any value above to recalculate instantly.
How to use the arithmetic sequence calculator
- 1Enter the first term of the sequence.
- 2Enter the common difference — subtract any term from the next one to find it.
- 3Enter which term number you want to find.
- 4Read the nth term, the sum of the first n terms and the explicit formula.
- 5Optionally enter a value to check whether it appears in the sequence.
Formula and method
In an arithmetic sequence each term is the previous term plus a fixed common difference d. Starting from a₁, you add d a total of n − 1 times to reach the nth term, which gives the explicit formula aₙ = a₁ + (n − 1)d.
The sum of the first n terms (an arithmetic series) is the number of terms times the average of the first and last terms. This is Gauss’s pairing trick: writing the series forwards and backwards and adding gives n copies of (a₁ + aₙ), so Sₙ = n(a₁ + aₙ) ÷ 2. To check whether a value x belongs to the sequence, solve x = a₁ + (k − 1)d for k; it is a term only if k is a positive whole number.
- a₁
- First term
- d
- Common difference between consecutive terms
- n
- Position of the term (1, 2, 3…)
- aₙ
- The nth term
- Sₙ
- Sum of the first n terms
Worked examples
3, 7, 11, 15… — the 10th term
a₁₀ = 3 + 9 × 4 = 39. The sum of the first 10 terms is 10 × (3 + 39) ÷ 2 = 210. Solving 43 = 3 + (k − 1) × 4 gives k = 11, so 43 is the 11th term.
Decreasing sequence: 100, 92.5, 85…
a₂₀ = 100 + 19 × (−7.5) = −42.5 and S₂₀ = 20 × (100 − 42.5) ÷ 2 = 575. For 50, k = (50 − 100) ÷ (−7.5) + 1 ≈ 7.67, which is not a whole number, so 50 is skipped.
Gauss’s sum 1 + 2 + … + 100
Pairing 1 + 100, 2 + 99 and so on gives 50 pairs of 101, so the sum is 100 × 101 ÷ 2 = 5,050.
Savings that grow by $0.50 a week
Saving $5 in week 1 and 50¢ more each week means week 12’s deposit is 5 + 11 × 0.5 = $10.50, and 12 weeks total 12 × (5 + 10.5) ÷ 2 = $93.
Frequently asked questions
How do you find the nth term of an arithmetic sequence?+
Use aₙ = a₁ + (n − 1)d. For 5, 8, 11, … the first term is 5 and d = 3, so the 20th term is 5 + 19 × 3 = 62.
How do you find the common difference?+
Subtract any term from the one after it: d = aₖ₊₁ − aₖ. If you know two non-consecutive terms, divide their difference by the gap in positions, e.g. (a₁₀ − a₄) ÷ 6.
What is the formula for the sum of an arithmetic series?+
Sₙ = n(a₁ + aₙ) ÷ 2, or equivalently Sₙ = n[2a₁ + (n − 1)d] ÷ 2 when you do not know the last term. It is the number of terms times the average of the first and last terms.
What is the difference between arithmetic and geometric sequences?+
An arithmetic sequence adds the same number each step (2, 5, 8, 11), so it grows linearly. A geometric sequence multiplies by the same ratio each step (2, 6, 18, 54), so it grows exponentially.
Can the common difference be negative or a fraction?+
Yes. A negative difference makes a decreasing sequence (20, 17, 14…) and a fractional difference gives steps like 1, 1.25, 1.5. The same formulas apply in every case.