Arithmetic Sequence Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/5/2026
The nth term of an arithmetic sequence is calculated as first term + (n−1) × common difference, and the sum of the first n terms as n × (2×first term + (n−1)×common difference) ÷ 2. For a first term of 3 and a common difference of 5, the 10th term is 48, for a sum of 255.
Explanation
An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a constant value, called the common difference, to the previous term: 3, 8, 13, 18, 23… is an arithmetic sequence with first term 3 and common difference 5, each term exceeding the previous one by 5. Rather than adding the common difference term by term up to the desired position, the direct formula uₙ = u₁ + (n−1) × r lets you calculate any term immediately, simply by multiplying the common difference by the number of 'steps' separating that term from the first (n−1 steps to reach the nth term). The sum of the first n terms is also calculated directly, without having to add each term one by one, thanks to a trick attributed to Carl Friedrich Gauss: by pairing the first and last term, the second and second-to-last, and so on, each pair gives exactly the same sum (2×u₁ + (n−1)×r), which just needs to be multiplied by the number of pairs (n ÷ 2). A negative common difference gives a decreasing rather than increasing sequence, as in the second example below; a common difference of zero gives a constant sequence, where each term simply equals the first. Arithmetic sequences model many real situations with constant linear change: a loan repaid with regularly increasing installments, seat numbering, or more generally any progression by equal steps.
Example: first term 3, common difference 5, 10th term
Inputs
First term (u₁): 3. Common difference (r): 5. Position sought (n): 10.
Calculation
u₁₀ = 3 + (10−1) × 5 = 3 + 45 = 48. Sum of the first 10 terms = 10 × (2×3 + 9×5) ÷ 2 = 10 × (6+45) ÷ 2 = 10 × 51 ÷ 2 = 255.
Result
The 10th term of this sequence is 48, and the sum of the first 10 terms is 255.
Frequently asked questions
How do I recognize that a sequence of numbers is arithmetic?
A sequence is arithmetic if the difference between two consecutive terms stays exactly the same throughout the sequence. To check, simply subtract each term from the next: if this result is constant (the common difference), the sequence is arithmetic. If this difference varies, the sequence is not arithmetic (it could, for example, be geometric, where it's the ratio between consecutive terms that stays constant, not their difference).
Where does the sum formula attributed to Gauss come from?
According to a famous anecdote, the mathematician Carl Friedrich Gauss, as a child, is said to have very quickly solved the exercise of adding all the numbers from 1 to 100 by noticing that pairing the first and last term (1+100=101), then the second and second-to-last (2+99=101), and so on, always gave the same sum per pair. By multiplying this constant sum by the number of pairs, he directly obtained the total result without having to add everything one by one — exactly the principle behind the general formula for the sum of an arithmetic sequence.
Does this formula also work for a common difference that isn't a whole number?
Yes, the common difference can be any real number, whole or not, positive, negative, or zero. The formula remains valid in all cases: a non-integer common difference simply gives a sequence whose terms aren't integers (unless the first term compensates exactly), without changing the calculation principle.