What this calculator does
An arithmetic sequence adds a fixed amount each step. Finding any term is straightforward, but the sum has a much more elegant shortcut than adding the terms one by one.
The shortcut is pairing. Gauss reportedly noticed as a schoolboy that pairing the first term with the last, the second with the second-last and so on gives identical totals, which turns a long addition into a single multiplication. Starting at 5 with a difference of 3, the first ten terms sum to 185.000.
The formula
The nth term adds the common difference n minus 1 times to the first term. The sum is n halves times the total of the first and last terms, which is what the pairing argument produces.
| Term | Meaning |
|---|---|
| Common difference | The fixed amount added at each step. |
| nth term | The value at position n in the sequence. |
| Partial sum | The total of the first n terms, also called an arithmetic series. |
The inputs explained
| Field | What to enter |
|---|---|
| First term (a₁) | The first term of the sequence. |
| Common difference (d) | The common difference. Negative values give a decreasing sequence. |
| Term number (n) | Which term to find, and how many terms to sum. |
When to use it
Finding a distant term
The formula jumps straight to any position without listing the sequence.
Summing a long series
The pairing shortcut avoids adding hundreds of terms individually.
Modelling linear growth
Anything increasing by a fixed amount per period is an arithmetic sequence.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do term and sum grow?
The same sequence evaluated at three lengths.
| Number of terms | nth term (aₙ) | Sum of first n terms (Sₙ) |
|---|---|---|
| n = 5 | 17.000 | 55.000 |
| n = 10 | 32.000 | 185.000 |
| n = 20 | 62.000 | 670.000 |
Questions
Why does the sum grow with the square of n?
Because you are adding n terms whose average size itself grows linearly with n. Multiplying a count that grows linearly by an average that grows linearly gives quadratic growth overall.
What is the Gauss pairing argument?
Pair the first term with the last, the second with the second-last, and so on. Every pair sums to the same total, because each step forward is matched by an equal step back, so the sum is that total times the number of pairs.
Does it work with a negative difference?
Yes, unchanged. A negative common difference gives a decreasing sequence, and both formulas handle it. The sum can then become negative once terms pass below zero.
How does this differ from a geometric sequence?
An arithmetic sequence adds a fixed amount each step; a geometric one multiplies by a fixed ratio. Arithmetic growth is linear, geometric growth is exponential, and the difference compounds enormously over many terms.
For the multiplying equivalent, see the geometric sequence calculator. For compounding in finance, see the compound interest calculator.