Key Idea: A geometric sequence multiplies by the same constant each step — the common ratio r.
Three things IB tests on this topic:
Find any term
- uₙ = u₁ × rⁿ⁻¹
- Gets one specific value
- e.g. "find the 8th term"
Find the sum
- Sₙ = u₁(1 − rⁿ) ÷ (1 − r) ← formula booklet form
- Sₙ = u₁(rⁿ − 1) ÷ (r − 1) ← equivalent form
- Adds up the first n terms
- S∞ = u₁ ÷ (1 − r) — Use u₁(1−rⁿ)÷(1−r), set rⁿ=0
Identify r and model
- r = u₂ ÷ u₁ (any consecutive pair)
- r > 1: growth | 0 < r < 1: decay
🌱 Growth vs decay
| Growth (r > 1) | Decay (0 < |r| < 1) | |
|---|---|---|
| What happens | Terms get bigger | Terms get smaller and approach 0 |
| Example | 6, 18, 54, 162, ... (r = 3) | 100, 50, 25, 12.5, ... (r = 0.5) |
| Real-world | Compound interest, population growth | Radioactive decay, depreciation |
| S∞ valid? | No — sum diverges to infinity | Yes — because |r| < 1 |
✏️ Worked examples
Find the nth term
A sequence: 4, 12, 36, 108, ... Find u₇.
Step by step:
Find r: 12 ÷ 4 = 3
Write the formula: uₙ = u₁ × rⁿ⁻¹
Substitute: u₇ = 4 × 3⁶
Calculate: 4 × 729 = 2916
Final answer:
u₇ = 2916
Find the sum of n terms
Same sequence: 4, 12, 36, ... Find S₅.
Step by step:
u₁ = 4, r = 3, n = 5
Formula: Sₙ = u₁(rⁿ − 1) ÷ (r − 1)
S₅ = 4 × (3⁵ − 1) ÷ (3 − 1) = 4 × 242 ÷ 2
Calculate: 4 × 121 = 484
Final answer:
S₅ = 484
Sum to infinity
Sequence: 8, 4, 2, 1, ... Does S∞ exist? Find it.
Step by step:
r = 4 ÷ 8 = 0.5 → |r| < 1 ✓ (S∞ exists)
Formula: S∞ = u₁ ÷ (1 − r)
S∞ = 8 ÷ (1 − 0.5) = 8 ÷ 0.5
S∞ = 16
Final answer:
S∞ = 16
Find r first — every problem starts here. Divide any term by the one before it. Exponent trap: The formula is u₁ × rⁿ⁻¹, not rⁿ. So the 1st term uses r⁰, the 2nd uses r¹, the 3rd uses r², and so on. Count up from zero. S∞ check: Always verify |r| < 1 before using S∞. Writing S∞ when |r| ≥ 1 will lose marks. Paper 2: For large powers use your GDC — type the full expression and press ENTER.