Find any term in a geometric sequence using aₙ = a₁·rⁿ⁻¹, and see the exponential pattern.
aₙ = a₁ · rⁿ⁻¹
Explicit formula: aₙ = a₁ · rⁿ⁻¹ gives the nth term directly without listing all previous terms.
Recursive formula: aₙ = r · aₙ₋₁. Each term is found by multiplying the previous term by the common ratio r.
Finding r: Divide any term by the one before it — r = a₂/a₁ = a₃/a₂ = aₙ/aₙ₋₁. The ratio is constant throughout.
Connection to exponential: Geometric sequences are the discrete version of exponential functions y = a · bˣ. The common ratio r plays the same role as the base b.
r determines how the sequence behaves
| Condition | Behavior | Example r |
|---|---|---|
| r > 1 | Growing (exponential growth) | r = 3 |
| 0 < r < 1 | Decaying toward 0 | r = 0.5 |
| r < 0 | Alternating signs | r = −2 |
| r = 1 | Constant (all terms equal a₁) | r = 1 |
| r = −1 | Oscillating between +a₁ and −a₁ | r = −1 |
| r = 0 | Degenerate (all terms 0 after a₁) | r = 0 |
Geometric sequences have a finite sum only when |r| < 1 (see Infinite Geometric Series).
One-on-one Algebra 2 tutoring builds intuition for sequences, series, and exponential patterns — we work through your actual homework and tests so the logic becomes second nature.