Arithmetic Sequence Calculator
Enter the first term, the common difference, and how many terms you want — get the nth term aₙ and the sum of the first n terms instantly.
Term and sum together
Enter a, d, and n and the calculator returns both the nth term aₙ = a + (n − 1)·d and the running sum Sₙ of the first n terms.
n is a whole count
The number of terms n must be a whole number of at least 1 — there is no zeroth or fractional term in a sequence.
What is an arithmetic sequence?
A list with a constant step
An arithmetic sequence is a list of numbers in which each term is obtained by adding a fixed amount — the common difference d — to the one before it: a, a+d, a+2d, … Because the step never changes, the whole sequence is pinned down by just two numbers, the first term a and the difference d. This calculator turns those two values, plus a term count n, into the nth term aₙ and the sum Sₙ of the first n terms without you writing the list out by hand.
Enter the first term, the common difference, and n to get the nth term and the sum of the first n terms.
The nth term comes from taking n − 1 steps of size d away from the first term, and the sum is the term count times the average of the first and last term.
aₙ = a + (n − 1) × d and Sₙ = n ÷ 2 × (2a + (n − 1) × d)First find the nth term by adding n − 1 copies of d to a. Then the sum follows from Sₙ = n/2·(2a + (n − 1)·d), which is just n multiplied by the average of the first term a and the last term aₙ. A positive d grows the sequence, a negative d shrinks it, and d = 0 keeps every term equal to a.
Suppose the sequence starts at a = 2 with a common difference d = 3, and you want the 10th term and the sum of the first 10 terms.
Find the nth term
a₁₀ = 2 + (10 − 1) × 3 = 2 + 27 = 29. The 10th term is 29.
Set up the sum
S₁₀ = 10 ÷ 2 × (2 × 2 + 9 × 3) = 5 × (4 + 27) = 5 × 31.
Compute the sum
S₁₀ = 5 × 31 = 155. The first ten terms add up to 155.
The calculator hands back two numbers, and each answers a different question. The nth term aₙ tells you the single value sitting at position n — for 2, 5, 8, … the tenth term is 29, the number you would land on after nine steps of three. The sum Sₙ tells you what all the terms from the first up to the nth add to: 155 for those same ten terms. A useful way to read the sum is as the number of terms times the average of the first and last term, since an arithmetic sequence is perfectly balanced around its middle. The sign of the common difference colours both outputs: a positive d means the terms and the running total climb, a negative d means they fall, and d = 0 means every term equals the first so the sum is simply n times that value.
The formulas are exact, but a couple of practical points are worth keeping in mind.
n must be a whole number of at least 1
The number of terms n counts positions in the list, so it must be a whole number of 1 or more — the calculator returns no result for a fractional or zero n. The first term and the common difference, by contrast, may be any numbers, positive or negative. This tool covers arithmetic sequences only, where a fixed amount is added each step; sequences that multiply by a constant ratio are geometric and need a different formula.