Arithmetic Sequence Calculator
Find any term and the running total of a sequence that grows by a fixed step.
The 10th term (a10)
48
a10 = 3 + (10 − 1) × 5
Sum of the first 10 terms
255
Using S = n/2 × (a₁ + aₙ) = 10/2 × (3 + 48)
The sequence starts
3, 8, 13, 18, 23, 28, 33, 38, …
How it works
An arithmetic sequence marches along by adding the same amount — the common difference — each step. Start at 3 with a difference of 5 and you get 3, 8, 13, 18, and so on.
To jump straight to any term without listing them all, use aₙ = a₁ + (n − 1) × d. The nth term is just the first term plus (n − 1) copies of the difference.
For the running total there's a neat shortcut: pair the first and last terms and you get S = n/2 × (a₁ + aₙ). Change any of the three inputs and both the nth term and the sum recalculate on the spot.
Frequently asked questions
What exactly is the common difference?
It's the fixed number you add to move from one term to the next. In 3, 8, 13, 18 the common difference is 5. A negative difference simply means the sequence counts down instead of up.
Why is the sum formula n/2 × (a₁ + aₙ)?
Line the sequence up forwards and backwards and every pair adds to the same total, a₁ + aₙ. There are n such pairings across the two copies, so dividing by two gives the sum — the trick Gauss famously used to add 1 through 100.
Can the difference be a decimal or negative?
Both work fine. A difference of 2.5 steps up by two and a half each time, and a difference of -4 steps down by four. The formulas don't care about the sign or whether the values are whole.