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Arithmetic Sequence Calculator

Find the nth term and sum of an arithmetic sequence.

How it is calculated

In an arithmetic sequence each term is the previous one plus a constant difference d. With first term a and number of terms n, the last (nth) term is an = a + (n − 1) × d and the sum of the first n terms is Sn = n ÷ 2 × (2a + (n − 1) × d), which equals the average of the first and last term times n. The number of terms is rounded to a whole number and must be at least 1. The first term and difference may be negative or decimal, and results are shown with four decimals.

Worked example

First term 5, common difference 3, 10 terms: 5, 8, 11, … Last term: 5 + (10 − 1) × 3 = 5 + 27 = 32.0000. Sum: 10 ÷ 2 × (2 × 5 + 9 × 3) = 5 × 37 = 185.0000. Check: the average of 5 and 32 is 18.5, and 18.5 × 10 = 185.

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Common mistakes

  • Using n instead of n − 1 in the nth-term formula, which overshoots by one difference. The first term already counts as term 1.
  • Entering the last term in the first-term field when you want the sum going backwards. Use a negative difference instead.
  • Confusing an arithmetic sequence (constant difference) with a geometric one (constant ratio). Doubling each time is geometric.
  • Entering a non-integer number of terms and expecting a partial term. The tool rounds to a whole number.

About this calculator

This calculator finds the nth term of an arithmetic sequence using aₙ = a₁ + (n−1)d, and the sum of its first n terms using Sₙ = n/2·(a₁ + aₙ), where a₁ is the first term and d is the common difference added at each step. These closed-form formulas let you jump straight to any term or partial sum without manually adding up a long list of numbers one at a time.

Algebra and pre-calculus students working through sequences-and-series homework, teachers building practice problems, and anyone modeling a linearly growing or shrinking pattern — such as stadium seating rows, savings that increase by a fixed amount each month, or a step-loaded staircase — uses this. Enter the first term, common difference, and n to get the nth term and the running sum.

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