The arithmetic nth-term formula
nth term = a₁ + (n−1)d, where a₁ is the first term and d is the common difference. The (n−1) accounts for the fact that the first term has already had the difference added zero times. For 2, 5, 8, 11, ... (a₁ = 2, d = 3), the 10th term is 2 + (10−1) × 3 = 2 + 27 = 29 — matching what you'd get by counting out all 10 terms by hand, but instantly for any term number, however large.
The geometric nth-term formula
nth term = a₁ × r^(n−1), where a₁ is the first term and r is the common ratio. For 2, 6, 18, 54, ... (a₁ = 2, r = 3), the 5th term is 2 × 3⁴ = 2 × 81 = 162. The exponent (n−1) plays the same role as in the arithmetic formula — the first term hasn't been multiplied by the ratio yet, the second term has been multiplied once, and so on.
A decreasing example for each
Both formulas work the same way when the sequence shrinks. An arithmetic sequence starting at 5 with a common difference of −2 runs 5, 3, 1, −1, −3, −5, −7, −9 — the 8th term is 5 + (8−1) × (−2) = 5 − 14 = −9. A geometric sequence starting at 100 with a common ratio of 0.5 runs 100, 50, 25, 12.5, 6.25, 3.125 — the 6th term is 100 × 0.5⁵ = 100 × 0.03125 = 3.125, halving every step rather than subtracting a fixed amount.
Finding the sum, not just one term
Both formulas have a matching sum formula for "add up the first n terms" rather than "find one specific term." For arithmetic sequences, Sum = n/2 × (2a₁ + (n−1)d) — for the 2, 5, 8, 11 example above, the sum of the first 10 terms is 155. For geometric sequences, Sum = a₁ × (rⁿ − 1) / (r − 1) for any ratio other than 1 — for the 2, 6, 18, 54 example, the sum of the first 5 terms is 242. Both calculators compute the nth term and this sum together, since they're usually wanted side by side.
Which formula to use
Use the arithmetic formula only when consecutive terms share a constant difference, and the geometric formula only when they share a constant ratio — see the companion article on telling arithmetic and geometric sequences apart for the quick test. A sequence that doesn't clearly pass either test, like Fibonacci, needs a different approach entirely.