How an arithmetic sequence is calculated
An arithmetic sequence adds the same common difference d to each term. The nth term extends that pattern directly, and the sum formula avoids adding every term one by one.
- a₁ — the first term
- d — the common difference between terms
- n — which term number to find (1, 2, 3, …)
- aₙ — the nth term
- Sₙ — the sum of the first n terms
More detail
Why the sum formula works
Sₙ pairs the first and last term (a₁ + aₙ), and every other pair from opposite ends of the sequence sums to the same value — that's why multiplying by n/2 gives the total without adding each term individually. It's the same trick used to sum 1 through 100 quickly.
Positive vs. negative common difference
A positive d makes the sequence increase (e.g. 1, 3, 5, 7, …), while a negative d makes it decrease (e.g. 10, 7, 4, 1, …). A common difference of 0 means every term equals a₁, a constant sequence.
Quick check. For a short sequence, you can sanity-check aₙ by counting: a₁, a₁+d, a₁+2d, … up to the nth term. If it matches the formula's output, your a₁, d, and n are entered correctly.
Frequently asked questions
What is the 5th term of 1, 3, 5, 7, …?
With a₁ = 1 and d = 2, the 5th term is aₙ = 1 + (5−1)×2 = 9. Enter a₁=1, d=2, n=5 above to see this along with the sum of the first 5 terms (25).
What is the sum of the first n terms of an arithmetic sequence?
Sₙ = n/2 × (2a₁ + (n−1)d). For a₁=1, d=2, n=5 this gives Sₙ = 5/2 × (2 + 8) = 25, matching the sum of 1+3+5+7+9.
Can the common difference be negative?
Yes — a negative d produces a decreasing sequence, such as a₁=10, d=−3 giving 10, 7, 4, 1. Both aₙ and Sₙ above handle negative differences the same way as positive ones.
Why must n be a whole number?
n represents a term's position in the sequence (1st, 2nd, 3rd, …), so it must be a positive integer. A value like 2.5 doesn't correspond to an actual term, which is why this calculator only accepts n ≥ 1.