Arithmetic sequences and series
An arithmetic sequence adds the same fixed amount — the common difference — to get from one term to the next: 2, 5, 8, 11 and so on. This calculator finds any term you ask for and the running total (the series) of the first n terms, given the starting value, the step, and how far out you want to go.
The two formulas
The nth term is a1 plus (n minus 1) times d — start, then take n minus 1 steps. The sum of the first n terms is n times the average of the first and last term, written n divided by 2 times (2a1 plus (n minus 1)d). Pairing terms from the ends is the classic trick behind that sum.
Sequence versus series
The sequence is the list of terms; the series is their sum. Telling them apart matters: a question about the 20th value wants the term formula, while total seats across rows of a theater wants the series formula.
Related tools
When each step multiplies instead of adds, use the geometric sequence calculator; for the famous add-the-last-two pattern, the Fibonacci calculator.
Worked example
Start at 2 with a common difference of 3. The 10th term is 2 + 9 times 3 = 29, and the first ten terms sum to 10 divided by 2 times (4 + 27) = 155.
FAQ
Can the common difference be negative?
Yes. A negative d makes a decreasing sequence, and the same formulas apply unchanged.
What if d is zero?
Every term equals the first, so the nth term is a1 and the sum is simply n times a1.