Math
Arithmetic Series Calculator
An arithmetic series adds terms that increase by the same amount each step, like 3 + 7 + 11 + 15.
Enter the first term, common difference, and how many terms (or the first and last terms) to get the exact sum, the last term, and a table of running totals.
Can be negative for a decreasing series.
Try an example
Result
Sum Sₙ
—
- Last term aₙ
- —
- Common difference
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- Average term
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- Partial sums
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What an arithmetic series is
Take an arithmetic sequence, where each term is the previous term plus a fixed difference d, and add the terms up. The running totals S₁, S₂, S₃, ... are the partial sums, and Sₙ is the sum of the first n terms.
Why the formula works
Write the series forward and backward and add the two copies. Every column sums to a₁ + aₙ, and there are n columns, so 2Sₙ = n(a₁ + aₙ). Divide by 2. For 1 + 2 + ... + 100 that is 100 × 101 / 2 = 5,050.
Famous arithmetic sums
| Series | a₁ | d | n | Sum |
|---|---|---|---|---|
| 1 + 2 + ... + 100 | 1 | 1 | 100 | 5,050 |
| 2 + 4 + ... + 100 | 2 | 2 | 50 | 2,550 |
| 1 + 3 + ... + 99 | 1 | 2 | 50 | 2,500 |
| 5 + 10 + ... + 500 | 5 | 5 | 100 | 25,250 |
Finding n when you know the last term
If the series runs from 7 to 91 in steps of 4, then n = (91 - 7)/4 + 1 = 22 terms, and the sum is 22 × (7 + 91)/2 = 1,078.
How to use it
- Choose whether you know the common difference or the last term.
- Enter the first term and the number of terms, plus the difference or last term.
- Read the sum, the last term, the average term, and the partial-sum table.
How to read the answer
The sum equals the number of terms times the average of the first and last term. That average is also the middle value of the series, so Sₙ = n × (middle value). A negative difference gives a decreasing series whose sum can be negative.
Common mistakes and edge cases
- Using aₙ = a₁ + n d instead of a₁ + (n - 1)d; the first term already counts as one.
- Applying this to a series that multiplies instead of adds; use the geometric series calculator.
- Mixing up the sequence's nth term with the series' sum of n terms.
Worked examples
3 + 7 + 11 + ... (10 terms)
First term 3, difference 4
Sum Sₙ
210
1 to 100
Gauss's classic
Sum Sₙ
5,050
From first and last
3 to 39 in 10 terms
Sum Sₙ
210
Decreasing
20 + 18 + ... (6 terms)
Sum Sₙ
90
Crosses zero
10 + 5 + 0 - 5 - 10
Sum Sₙ
0
Decimal difference
0.5 + 1.25 + 2 + ... (8 terms)
Sum Sₙ
25
Single term
n = 1 gives just the first term
Sum Sₙ
7
Impossible
One term but different first and last
Sum Sₙ
Error
Frequently asked questions
What is the arithmetic series formula?+
Sₙ = n(a₁ + aₙ)/2, or equivalently Sₙ = n/2 × [2a₁ + (n - 1)d] when you know the difference instead of the last term.
What is the sum of 1 to 100?+
5,050. Pair 1 with 100, 2 with 99, and so on: 50 pairs of 101.
How do I find the number of terms?+
n = (aₙ - a₁)/d + 1. Add 1 because both endpoints are included.
Can the sum be negative?+
Yes, if enough terms are negative. 10 + 5 + 0 - 5 - 10 - 15 sums to -15.
What is the difference between an arithmetic sequence and an arithmetic series?+
The sequence is the list of terms; the series is their sum. This page sums; the arithmetic sequence calculator finds individual terms.
Does an arithmetic series converge?+
No. Unless d = 0 and a₁ = 0, the terms never shrink to zero, so the infinite sum does not exist.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026