Finds a specific term and the running sum of a sequence that changes by the same fixed amount at each step.
How it works
The nth term is found by adding the common difference (n−1) times to the first term; the sum uses a formula that averages the first and nth terms across all n terms.
What this does not include
This does not include geometric sequences, which multiply by a fixed ratio rather than adding a fixed difference — a different type of sequence entirely.
How to use this calculator
- Enter the first term, the common difference, and which term number you want.
A worked example
First term 2, common difference 3, 5th term: nth term = a1 + (n−1)d = 2 + 4×3 = 14; sum of first 5 terms = 40.
First term 10, common difference −2, 6th term: nth term = 0; sum of first 6 terms = 30.
What the variables mean
| Variable | Meaning |
|---|---|
| First term (a1) | The sequence’s starting value |
| Common difference (d) | The fixed amount added each step |
| n | Which term number to find |
Edge cases worth knowing
A negative common difference produces a decreasing sequence, not an error — the second example shows a sequence counting down by 2 each term, reaching exactly 0 at the 6th term.
n must be at least 1 — there’s no “zeroth term” in a standard arithmetic sequence, so the calculator declines to show a result for n=0.
Frequently asked questions
What makes a sequence “arithmetic”?
Each term differs from the one before it by the same fixed amount, called the common difference.
Can the common difference be negative?
Yes — a negative common difference simply means the sequence decreases instead of increases with each step.
What’s a real-world example of an arithmetic sequence?
Seating rows in a theater that each add a fixed number of extra seats, or a savings plan that adds the same fixed amount each month (ignoring interest).
