You enter a first term, choose arithmetic or geometric, provide the common difference (d) or common ratio (r), and specify how many terms. For arithmetic sequences, the nth term is a₁ + (n−1)d and the sum is (n/2)(2a₁ + (n−1)d). For geometric sequences, the nth term is a₁ × r^(n−1) and the sum is a₁(1 − r^n)/(1 − r). It assumes exact values with no rounding until display.
An arithmetic sequence adds a constant amount each step (2, 5, 8, 11 — adding 3). A geometric sequence multiplies by a constant each step (3, 6, 12, 24 — multiplying by 2). Arithmetic grows linearly, geometric grows exponentially.
Every term equals the first term, and the sum is simply the first term times the number of terms. The calculator handles this as a special case to avoid dividing by zero in the geometric sum formula.
Yes. A negative common difference creates a decreasing arithmetic sequence. A negative ratio creates an alternating-sign geometric sequence (e.g., 1, −2, 4, −8). Both are mathematically valid and the formulas still apply.
Arithmetic sequences model equal payments, uniform motion, and evenly spaced measurements. Geometric sequences model compound interest, population growth, radioactive decay, and any process where a quantity changes by a fixed percentage each period.
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.