Arithmetic Sequence Calculator
Calculate the nth term, sum of n terms, or position of a term in an arithmetic sequence. Ideal for students and professionals working with linearly changing data.
About this calculator
An arithmetic sequence is a list of numbers where each term increases or decreases by a fixed amount called the common difference (d). The nth term is given by: aₙ = a₁ + (n − 1) × d, where a₁ is the first term and n is the position. The sum of the first n terms is: Sₙ = (n / 2) × (2a₁ + (n − 1) × d). This can also be written as Sₙ = n × (a₁ + aₙ) / 2, which is the number of terms times the average of the first and last term. If you know a specific term value and want its position, rearrange the nth-term formula: n = (aₙ − a₁) / d + 1. Arithmetic sequences model real-world patterns like evenly spaced salaries, seat rows, or loan repayments.
How to use
Suppose a₁ = 3, d = 5, and you want the 10th term and the sum of the first 10 terms. Nth term: a₁₀ = 3 + (10 − 1) × 5 = 3 + 45 = 48. Sum of 10 terms: S₁₀ = (10 / 2) × (2 × 3 + (10 − 1) × 5) = 5 × (6 + 45) = 5 × 51 = 255. Enter first_term = 3, common_difference = 5, term_position = 10, and select the desired calculation type to get your result instantly.
Frequently asked questions
What is the difference between an arithmetic sequence and an arithmetic series?
An arithmetic sequence is the ordered list of terms: a₁, a₂, a₃, … each separated by the common difference d. An arithmetic series is the sum of some or all of those terms. For example, the sequence 2, 5, 8, 11 becomes the series 2 + 5 + 8 + 11 = 26 when you add them. This calculator can compute both the individual terms and the cumulative sum, so it handles both concepts.
How do I find the common difference of an arithmetic sequence?
The common difference d is found by subtracting any term from the term that immediately follows it: d = aₙ₊₁ − aₙ. For example, in the sequence 7, 11, 15, 19, the common difference is 11 − 7 = 4. It is constant throughout the sequence, so checking one pair is sufficient. If the differences between consecutive terms are not equal, the sequence is not arithmetic.
When would I use the sum formula for an arithmetic sequence in real life?
The sum formula is useful any time you need to total a linearly growing or shrinking quantity over multiple periods. Classic examples include calculating the total pay over a career with annual raises, totalling seat counts across rows in a tiered stadium, or adding up distances when acceleration is constant. It also appears in finance for simple interest accumulation. Instead of adding every term individually, the formula gives you the answer in one step.