Arithmetic Series Sum Calculator
Add up the terms of an arithmetic sequence — enter the first term, common difference and number of terms to get the series sum and last term.
How it works
nth term aₙ = a₁ + (n − 1)·d · Sum Sₙ = n ÷ 2 · (a₁ + aₙ) = n ÷ 2 · [2a₁ + (n − 1)·d]
An arithmetic series adds equal-step terms: 2, 5, 8, 11, … The trick Gauss famously used is to pair the first and last terms, the second and second-to-last, and so on — every pair sums to a₁ + aₙ, and there are n ÷ 2 pairs. That gives the sum instantly as n ÷ 2 times (a₁ + aₙ), no term-by-term addition required. Equivalently, the sum is n times the average of the first and last terms.
Worked example
For a first term of 2, common difference 3 and 10 terms: the last term is a₁₀ = 2 + 9 × 3 = 29. The sum is 10 ÷ 2 × (2 + 29) = 5 × 31 = 155. Check the average: 155 ÷ 10 = 15.5, exactly halfway between the first term 2 and the last term 29.
Sources: Arithmetic progression — Wikipedia
Frequently asked questions
What is the difference between a sequence and a series?
A sequence is the ordered list of terms (2, 5, 8, …); a series is their running sum (2, 7, 15, …). This tool reports the series sum after n terms, along with the value of the nth term itself.
Can the common difference be negative or a decimal?
Yes. A negative difference makes a decreasing series, and decimals work fine — the formula only requires that each step adds the same fixed amount. Enter any value for d.
Why is the sum n times the average term?
Because the terms are evenly spaced, their mean is exactly the average of the first and last, (a₁ + aₙ) ÷ 2. Multiplying that mean by the count n reproduces the sum, which is why Sₙ = n(a₁ + aₙ) ÷ 2.
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This calculator is for educational purposes. Double-check important results before acting on them.