Find specific terms and lists in arithmetic or geometric sequences.
Overview
Sequences are ordered lists following a rule. Arithmetic sequences add a constant difference each term; geometric sequences multiply by a constant ratio. Recognizing which type you have determines the correct nth-term formula.
Arithmetic nth term: aₙ = a₁ + (n−1)d. Geometric nth term: aₙ = a₁ × r^(n−1). Sum formulas extend to series calculations for loan schedules, compound growth discrete steps, and pattern puzzles.
Misidentifying a sequence type produces wrong extrapolations. Plot terms versus n: roughly linear spacing suggests arithmetic; exponential curvature suggests geometric.
Use this calculator to generate tables for homework, forecast simple recurring increments, or verify competition math pattern problems before proving a closed form algebraically.
Steps
Select arithmetic or geometric.
Provide a₁.
Use d for arithmetic or r for geometric.
Enter n for the nth term.
Optionally show first k terms.
Check differences or ratios between consecutive terms.
Math
Arithmetic: aₙ = a₁ + (n−1)d; Geometric: aₙ = a₁ r^(n−1)
Each term applies the same additive step (arithmetic) or multiplicative step (geometric) relative to the prior term.
Worked cases
4 + 9×3 = 31
a₁₀ = 31.
5×2^5 = 160
a₆ = 160.
Constant difference 3
Arithmetic with d=3.
Why use this
Questions