Arithmetic Sequence Calculator
Enter the first term, the common difference, and how many terms you want. The calculator lists every term, the nth term aₙ, and the running total Sₙ.
nth term (a₁₀)
29
- Sum of 10 terms (Sₙ)
- 155
- First term (a₁)
- 2
- Terms
- 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
aₙ = a₁ + (n − 1) d. The partial sum is Sₙ = n/2 × (a₁ + aₙ), equivalent to n/2 × (2a₁ + (n − 1) d). Source: OpenStax Algebra & Trigonometry, ch. 9.
How to use this calculator
Type the first term a₁ (any real number), the common difference d (the fixed amount added to each term — negative for a decreasing sequence), and the number of terms n (a whole number from 1 to 1000). The calculator prints the full list of terms, the value of the nth term, and the partial sum Sₙ.
How the calculation works
An arithmetic sequence is a list of numbers in which the gap between consecutive terms is the same constant d. The formula for the nth term is aₙ = a₁ + (n − 1) d, and the partial sum of the first n terms is Sₙ = n/2 × (a₁ + aₙ), equivalent to n/2 × (2a₁ + (n − 1) d). Both formulas come from pairing terms equidistant from the ends — a method usually credited to a young Gauss. The calculator computes every term in one pass and uses the closed-form sum so even 1,000-term sequences resolve instantly. Source: OpenStax Algebra & Trigonometry, ch. 9.
Worked example
Take a₁ = 2 and d = 3 with n = 10 terms. The sequence is 2, 5, 8, 11, 14, 17, 20, 23, 26, 29. The tenth term is a₁₀ = 2 + (10 − 1) × 3 = 2 + 27 = 29. The sum is S₁₀ = 10/2 × (2 + 29) = 5 × 31 = 155. Cross-checking with the second sum form: 10/2 × (2 × 2 + 9 × 3) = 5 × 31 = 155 — matches.
Frequently asked questions
What is the difference between an arithmetic sequence and a geometric sequence?
An arithmetic sequence has a constant difference between terms — you add the same amount each step. A geometric sequence has a constant ratio — you multiply by the same amount each step. So 3, 7, 11, 15 is arithmetic (d = 4), but 3, 6, 12, 24 is geometric (r = 2). If you need the geometric version, use the geometric sequence calculator instead.
Can the common difference be negative or zero?
Yes. A negative common difference produces a strictly decreasing sequence: a₁ = 100, d = −7 gives 100, 93, 86, 79, … A common difference of zero produces a constant sequence — every term equals a₁. In that case Sₙ simplifies to n × a₁ because the formula n/2 × (a₁ + aₙ) collapses to n/2 × 2a₁.
What is the formula for the nth term?
aₙ = a₁ + (n − 1) d. The reason for the (n − 1) and not n is that the first term itself is a₁ + 0 × d — you have only added d zero times by the time you reach term one. By the time you reach term n you have added d a total of (n − 1) times.
How is the sum formula derived?
Write the sum forwards: Sₙ = a₁ + a₂ + … + aₙ. Write it again backwards: Sₙ = aₙ + aₙ₋₁ + … + a₁. Add the two row by row: every pair sums to (a₁ + aₙ), and there are n pairs, so 2 Sₙ = n × (a₁ + aₙ). Divide by two to get Sₙ = n/2 × (a₁ + aₙ). Substituting aₙ = a₁ + (n − 1) d gives the second form, Sₙ = n/2 × (2 a₁ + (n − 1) d).
Does the calculator work with decimals and fractions?
Yes — a₁ and d can be any real number, including decimals like 0.5, −2.75, or 3.14. The number of terms n must be a whole number because you cannot have half a term in a sequence; the calculator rejects fractional n. Very large terms (above 10¹²) are shown in scientific notation to keep the output readable.
Is there a limit on the number of terms?
The calculator caps n at 1,000 to keep the page responsive — listing more than that becomes unreadable in a browser. For sequences longer than 1,000 terms, the nth term and the partial sum can still be computed by hand from the closed-form formulas above without enumerating every value: that is exactly the point of the closed-form sum.