Geometric sequences
A geometric sequence multiplies by the same number each time: the common ratio, r.
| Sequence | a | r |
|---|---|---|
| 2, 6, 18, 54, … | 2 | 3 |
| 80, 40, 20, 10, … | 80 | 1/2 |
| 5, −10, 20, −40, … | 5 | −2 |
To check: divide each term by the one before it. If the result is always the same, it’s geometric.
The general term
tₙ = a × r⁽ⁿ⁻¹⁾
This is the general term: the first term a multiplied by r, (n − 1) times.
Example: 2, 6, 18, … Find t₈.
t₈ = 2 × 3⁷ = 2 × 2,187 = 4,374
Growth and decay
| r | Behaviour |
|---|---|
| r > 1 | grows quickly: exponential growth |
| 0 < r < 1 | shrinks toward 0: exponential decay |
| r < 0 | alternates between positive and negative |
Real examples:
- Bacteria doubling every hour (r = 2)
- A ball bouncing to 60% of its height (r = 0.6)
- Compound interest at 5% (r = 1.05)
Geometric series
The sum of the first n terms (when r ≠ 1) is a geometric series:
Sₙ = a(rⁿ − 1)/(r − 1)
Example: 3 + 6 + 12 + … (8 terms)
S₈ = 3(2⁸ − 1)/(2 − 1) = 3 × 255 = 765
The famous chessboard
A legend says a king offered a reward: 1 grain of rice on the first square of a chessboard, 2 on the second, 4 on the third, doubling each time. Total for 64 squares:
S₆₄ = 2⁶⁴ − 1 ≈ 1.8 × 10¹⁹ grains, far more than all the rice ever grown.
Infinite geometric series
If −1 < r < 1, the terms get so small that the sum approaches a fixed value. An infinite series like this has the sum:
S = a/(1 − r)
Example: 1/2 + 1/4 + 1/8 + … → a = 1/2, r = 1/2 → S = (1/2)/(1/2) = 1.
Worked examples
Example 1: Find t₆ for 81, 27, 9, … → r = 1/3 → 81 × (1/3)⁵ = 81/243 = 1/3.
Example 2: A ball dropped from 10 m bounces to 70% of its height each time. Height after the 4th bounce? 10 × 0.7⁴ ≈ 2.4 m.
Example 3: Sum the first 6 terms of 5, 10, 20, … → 5(2⁶ − 1) = 315.
Common mistakes
- Using r instead of r⁽ⁿ⁻¹⁾, or n instead of n − 1.
- Confusing arithmetic (add) and geometric (multiply).
- Using the infinite sum formula when |r| ≥ 1.
Try it yourself
- Find r and t₇ for 4, 12, 36, …
- Find t₅ for 1,000, 100, 10, …
- Sum the first 5 terms of 2, 8, 32, …
- Sum the infinite series 9 + 3 + 1 + …
- Is 3, 6, 9, 12 geometric?
Answers: 1) r = 3, t₇ = 2,916 2) 0.1 3) 682 4) 13.5 5) No, it’s arithmetic.
Quick recap
- A geometric sequence has a common ratio r.
- General term: tₙ = ar⁽ⁿ⁻¹⁾.
- Geometric series: Sₙ = a(rⁿ − 1)/(r − 1); infinite sum a/(1 − r) when |r| < 1.