Arithmetic series sum calculator

Pick whether you know the ends of the list or the step. The sequence 3, 6, 9, 12, 15, 18, 21 has step 3 and sum 84. In first term and step mode: 2, 5, 8, 11 adds to 26.

An arithmetic sequence adds (or subtracts) the same amount from term to term - that constant step is the common difference. You get two input modes: when you know the ends of the list, or when you know the step. The sum is the same; only the starting data changes.

Input

What do you know?

You know the first and last terms. We will find the step between them and the sum.

Result

Fill in the fields to see the sum.

What it means

An arithmetic sequence looks like stairs with equal rises. Example: 3, 6, 9, 12, 15, 18, 21. You add 3 from term to term. First term 3, last 21, seven terms, step 3. Sum 3+6+9+12+15+18+21. Shortcut: average of the ends (3+21)/2 = 12, times 7 is 84.

That is why there are two modes. Sometimes a worksheet gives the ends. Sometimes it gives the step and the count, and you do not have the last term yet. Both reach the same sum. 2, 5, 8, 11: step 3, n = 4, last term 11, sum 26.

Pick What do you know?: First and last term, or First term and step. Always type First term and Number of terms (a whole number greater than zero). In the first mode add Last term. In the second add Step (common difference). Click Calculate.

The geometric sum 2+6+18+54 = 80 multiplies by a ratio, another calculator. A mean of a list with no equal-step assumption lives on arithmetic mean.

Without the fields the mode needs, nothing computes. n = 7.5 does not run. A negative step means the sequence falls.

1+2+3+…+10 = 55. 10+20+30+40 = 100. First 5, step −2, n = 4: 5+3+1+(−1) = 8.

Two ways to enter data

  • First and last term - when you know the start and end of the list and how many terms there are. The calculator finds the step.
  • First term and step - when you know the start and how much each next term rises or falls. The calculator finds the last term, then the sum.

Both ways use the same idea: sum = number of terms × (first + last) ÷ 2.

Where it helps

  • School work on arithmetic sequences - with ends or with the common difference.
  • Quickly adding many numbers that rise or fall by a fixed step.
  • Checking homework without writing out the whole list.
  • Simple models with a steady day-by-day increase.

How to use

  1. Pick What do you know?: First and last term, or First term and step.
  2. Type the first term and the number of terms, a whole number greater than zero, for example 3 and 7.
  3. In ends mode type the last term, for example 21. In step mode type the step, for example 3.
  4. Click Calculate. 3 through 21 in seven terms is 84. 2, step 3, four terms is 26.
  5. A times-three ratio lives on the geometric series sum. The calculator adds a fixed step.

S = 84 from 3 to 21 in 7 terms

The sequence 3, 6, 9, 12, 15, 18, 21 has step 3 and sum 84. In first term and step mode: 2, 5, 8, 11 adds to 26.

S
Sum n × (first + last) / 2. Seven terms from 3 to 21: mean 12 times 7 = 84. Four terms from 2 with step 3: 26.
n
Term count, a whole number. 7 with 3 and 21. 4 with start 2 and step 3. n = 7.5 does not run.
step
The constant difference d. 3 in 3, 6, …, 21. 3 in 2, 5, 8, 11. A negative step means the sequence falls.

How the result is computed

When you know first and last: sum = n × (first + last) ÷ 2. Step = (last − first) ÷ (n − 1) when n is greater than 1.

When you know first and step: last term = first + (n − 1) × step. Then the same sum: n × (first + last) ÷ 2.

The result also shows a short list of terms (up to eight) so you can check the sequence quickly.

Examples

Example 1

  • Mode: first and last
  • First: 3
  • Last: 21
  • Terms: 7

84

Terms: 3, 6, 9, 12, 15, 18, 21. Step is 3. Sum: 84.

Example 2

  • Mode: first and step
  • First: 2
  • Step: 3
  • Terms: 4

26

Terms: 2, 5, 8, 11. Last term becomes 11. Sum: 26.

Example 3

  • Mode: first and last
  • First: 20
  • Last: 5
  • Terms: 4

50

Terms: 20, 15, 10, 5. Step is −5. Sum: 50.

Related calculators

Geometric series sum

Geometric series sum calculator: the first term, the ratio, and the term count.

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Absolute value calculator

Absolute value calculator: a number, plus or minus.

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FAQ

What is the sum 3+6+…+21?

84. Seven terms with step 3: the end average is 12, and 12 × 7 = 84.

What is 2+5+8+11?

26. First term and step mode: start 2, step 3, n = 4, last term comes out 11.

What is the sum formula?

S = n × (first + last) / 2. Or n/2 × (2a + (n−1)d) when you know the step d.

How do the modes differ?

Ends: you type first and last. Step: you type first and the difference; the calculator finds the last term.

Can n be 7.5?

No. The number of terms must be a whole number greater than zero.

What is 1+2+…+10?

55. Ten terms from 1 to 10: the end average is 5.5, and 5.5 × 10 = 55.

Can the step be negative?

Yes. The sequence falls. 5, 3, 1, −1 with n = 4 adds to 8.

How is this different from a geometric series?

Here a fixed plus. There a fixed ratio. 2+6+18+54 = 80 is on that page.

What is 10+20+30+40?

100. Four terms with step 10: the end average is 25, and 25 × 4 = 100.

Does the mean of 2 4 6 belong here?

No. A mean of any list lives on arithmetic mean. Here you assume an equal step.

Knowledge sources

The calculator computes the same formula as the definition below.

Page updated in 2026.