Find the nth term, sum, and common difference of an arithmetic sequence from the first term, common difference, and count — with formula and steps.
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An arithmetic sequence is a list of numbers where each term increases or decreases by the same amount — the common difference. This calculator finds any term and the sum of a sequence from three inputs: the first term, the common difference, and how many terms you want. It shows the nth-term and sum formulas with your numbers plugged in, lists the terms, graphs the progression, and works step by step, so it's as useful for learning the formulas as for getting a quick answer.
An arithmetic sequence (or arithmetic progression) is a sequence in which the difference between consecutive terms is constant. That constant is the common difference, d. Starting from the first term a₁, each term adds d: a₁, a₁+d, a₁+2d, and so on. The nth term is found with aₙ = a₁ + (n − 1)d, and the sum of the first n terms is Sₙ = n/2 × (a₁ + aₙ). When d is positive the sequence increases; when d is negative it decreases; when d is zero every term is the same.
nth Term & Sum
Solve nth-term and sum problems with full working shown.
See aₙ = a₁ + (n−1)d and Sₙ applied to real numbers.
Jump straight to the 500th term without listing them all.
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Enter a first term and difference to generate and visualize the sequence.
Practice arithmetic-progression questions for algebra exams.
Get the 10th, 100th, or 1,000th term without writing out the whole sequence.
Compute the sum of the first n terms in one step with the series formula.
Both the explicit and recursive formulas are shown with your numbers, not just the answer.
A line chart shows the sequence's steady linear pattern at a glance.
Step-by-step working makes it easy to check answers and learn the method.
The nth term is aₙ = a₁ + (n − 1)d, where a₁ is the first term, d is the common difference, and n is the term number. For example, with a₁ = 3 and d = 4, the 10th term is 3 + (10 − 1)×4 = 39.
Use Sₙ = n/2 × (a₁ + aₙ), or equivalently Sₙ = n/2 × [2a₁ + (n − 1)d]. For the sequence 3, 7, 11, … the sum of the first 10 terms is 10/2 × (3 + 39) = 210.
The common difference, d, is the constant amount added to each term to get the next one. Find it by subtracting any term from the term after it: d = a₂ − a₁. A positive d means the sequence increases; a negative d means it decreases.
Here a₁ = 3 and d = 4, so the 10th term is a₁₀ = 3 + (10 − 1)×4 = 3 + 36 = 39. The sum of the first 10 terms is 210.
An arithmetic sequence adds a constant (the common difference) each term, giving a straight-line pattern. A geometric sequence multiplies by a constant (the common ratio) each term, giving exponential growth or decay. This calculator handles the arithmetic case.
Check whether the difference between consecutive terms is always the same. If a₂ − a₁ equals a₃ − a₂ equals a₄ − a₃, and so on, the sequence is arithmetic and that constant is the common difference.