An arithmetic sequence is one of the simplest and most useful patterns in mathematics: start with a number, then keep adding (or subtracting) the same amount to get the next. This calculator finds any term in the sequence directly, totals up a run of terms, and lists them out so you can see the pattern for yourself.
How the nth term formula works
Rather than walking through every term one by one, the formula aₙ = a₁ + (n − 1)d jumps straight to any position. It works because getting from the first term to the nth term always takes exactly (n − 1) steps of size d — the first term itself needs zero steps, the second term needs one step, and so on. This makes it easy to find, say, the 500th term without ever writing out the first 499.
Why the sum formula pairs first and last terms
The sum formula Sₙ = n/2 · (a₁ + aₙ) relies on a classic trick: write the sum forwards and backwards, then add the two copies together. Each pair (first + last, second + second-to-last, and so on) adds up to the same total, a₁ + aₙ, and there are n such pairs when you count both copies — so dividing by 2 gives the actual sum. This is the same insight famously attributed to a young Gauss summing 1 through 100.
Inputs, limits, and edge cases
a₁ and d accept any real number, including negatives and decimals — a negative d simply produces a decreasing sequence, and d = 0 produces a constant one. n must be a whole number of 1 or greater; this calculator caps n at 1,000,000 to keep the math well-defined, and the displayed term list is capped at 500 terms so the page stays responsive, though aₙ and Sₙ are still computed exactly for the full n you entered.