An arithmetic sequence (or progression) adds a fixed common difference d between terms. From the first term a, the nth term is aₙ = a + (n−1)d, and the sum of the first n terms is Sₙ = n/2 × (a + aₙ) — the count times the average of the first and last terms.
Reviewed: June 20, 2026 · Author: Naveen P N, Founder — AI Calculator · Verified against: the arithmetic nth-term and series-sum formulas, recomputed in code.
The formulas
The nth term steps forward from the first by (n − 1) lots of d. The sum is the number of terms times their average, and because the terms are evenly spaced, that average is just the midpoint of the first and last — which is why Sₙ = n/2 × (a + aₙ). The two sum forms are the same; the second avoids computing aₙ first.
Worked examples
a = 2, d = 3, n = 10:
A decreasing one, a = 5, d = −2, n = 8:
The Gauss sum, a = 1, d = 1, n = 100:
So the first runs to a 10th term of 29 summing to 155, the decreasing one reaches −9 summing to −16, and adding 1 through 100 famously gives 5050.
Frequently Asked Questions
Terms that differ by a constant d. 2, 5, 8, 11 has d = 3. Also called an arithmetic progression.
aₙ = a + (n−1)d. a = 2, d = 3 → a₁₀ = 2 + 27 = 29.
Sₙ = n/2 × (a + aₙ). For a=2,d=3,n=10: 10/2 × (2+29) = 155.
Yes — a decreasing sequence. a=5, d=−2 → 5,3,1,−1,… with a₈ = −9.
Arithmetic adds a difference; geometric multiplies by a ratio. 2,5,8 vs 2,6,18.