Understanding the sequence preview
The sign of the common difference determines whether the sequence increases, decreases, or stays constant, as shown below for a first term of 3.
| Common difference (d) | Behavior | First 5 terms (a₁ = 3) |
|---|---|---|
| d > 0 (e.g. d = 4) | Increasing sequence | 3, 7, 11, 15, 19 |
| d = 0 | Constant sequence — every term equals a₁ | 3, 3, 3, 3, 3 |
| d < 0 (e.g. d = −4) | Decreasing sequence | 3, −1, −5, −9, −13 |
- The sequence preview shows up to the first 8 terms; for n greater than 8, an ellipsis (…) indicates the sequence continues beyond what is displayed, though the nth term and sum are still computed exactly for the full n you entered.
- Unlike a geometric sequence, an arithmetic sequence has no upper bound on how large or small terms can become as n increases (unless d = 0), and there is no finite 'sum to infinity' — the partial sum Sₙ grows without bound as n increases (for d ≠ 0).
What is an arithmetic sequence?
An arithmetic sequence (or arithmetic progression) is an ordered list of numbers in which the difference between each term and the one before it is always the same constant, called the common difference (d). For example, 3, 7, 11, 15, 19 is an arithmetic sequence with first term 3 and common difference 4, because each term is 4 more than the previous one.
The common difference can be positive (an increasing sequence), negative (a decreasing sequence), or zero (a constant sequence in which every term equals the first term). Arithmetic sequences model any process that changes by the same fixed amount at each step, such as a savings plan with equal fixed deposits, seating rows with a constant number of additional seats, or a countdown timer decrementing by a fixed interval.
Two related quantities are commonly asked about an arithmetic sequence: the value of a specific term (the nth term), and the sum of a run of consecutive terms (a partial sum, sometimes called an arithmetic series). This calculator computes both from the same three inputs.
How to use this arithmetic sequence calculator
- Enter the first term of the sequence (a₁) — the starting value.
- Enter the common difference (d) — the fixed amount added to each term to get the next one. Use a negative number for a decreasing sequence.
- Enter the number of terms (n) you want to find the nth term and sum for.
- Read the nth term, the sum of the first n terms, and a preview showing up to the first 8 terms of the sequence.
The arithmetic sequence formulas
The nth term of an arithmetic sequence is found by starting at the first term and adding the common difference (n − 1) times, since the first term itself requires zero additions: aₙ = a₁ + (n − 1)d. Worked example with a₁ = 3, d = 4, n = 10: a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39.
The sum of the first n terms uses the fact that pairing the first and last terms, the second and second-to-last terms, and so on, always gives the same pair-sum (a₁ + aₙ). This yields the closed-form sum Sₙ = (n/2) × (2a₁ + (n − 1)d), equivalently Sₙ = (n/2) × (a₁ + aₙ). Worked example: S₁₀ = (10/2) × (2 × 3 + 9 × 4) = 5 × (6 + 36) = 5 × 42 = 210.
This sum formula is attributed in mathematical folklore to the young Carl Friedrich Gauss, who is said to have summed the integers 1 through 100 by pairing 1+100, 2+99, and so on, giving 50 pairs each summing to 101, for a total of 5050 — the same pairing logic underlies the general formula used here.
Common mistakes
- Using n instead of (n − 1) in the nth-term formula — the first term (n = 1) requires zero additions of d, not one, since aₙ = a₁ + (n − 1)d.
- Confusing the common difference with a common ratio — arithmetic sequences add a fixed amount each step; geometric sequences multiply by a fixed factor instead.
- Assuming the sum formula requires listing every term — the closed-form formula Sₙ = (n/2)(2a₁ + (n − 1)d) computes the sum directly without needing to add each term individually.
- Entering a non-whole number of terms — n must be a positive whole number representing a term position (1st, 2nd, 3rd, …), not a fractional count.
Frequently asked questions
How do you find the nth term of an arithmetic sequence?
Use the formula aₙ = a₁ + (n − 1)d, where a₁ is the first term, d is the common difference, and n is the term number you want. For a₁ = 3, d = 4, and n = 10: a₁₀ = 3 + (10 − 1) × 4 = 3 + 36 = 39.
How do you find the sum of an arithmetic sequence?
Use the formula Sₙ = (n/2) × (2a₁ + (n − 1)d), which sums the first n terms without needing to add each one individually. For a₁ = 3, d = 4, n = 10: S₁₀ = (10/2) × (6 + 36) = 5 × 42 = 210.
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence has a constant difference between consecutive terms (each term = previous term + d). A geometric sequence has a constant ratio between consecutive terms (each term = previous term × r). Arithmetic sequences grow linearly; geometric sequences grow (or shrink) exponentially when the ratio's magnitude is not 1.
Can the common difference be negative or zero?
Yes. A negative common difference produces a decreasing sequence (e.g. 10, 7, 4, 1, −2 with d = −3). A common difference of zero produces a constant sequence in which every term equals the first term. Both are valid arithmetic sequences.
How do you find the common difference from two terms?
Subtract any term from the term that immediately follows it: d = aₙ₊₁ − aₙ. If you only know two non-consecutive terms, aₘ and aₙ, use d = (aₙ − aₘ) ÷ (n − m).
References
- NIST Digital Library of Mathematical Functions (DLMF), §1.2 Elementary Algebra: Arithmetic Progression. dlmf.nist.gov.
- Rosen KH. Discrete Mathematics and Its Applications. 8th ed. McGraw-Hill, 2018. (Sequences and series.)
- Weisstein EW. 'Arithmetic Series.' MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.