Geometric Sequences - Common Ratio and Exponential Growth
A geometric sequence is a list of numbers in which each term is obtained from the previous one by multiplying by a fixed amount called the common ratio, usually written r. Geometric sequences grow (or shrink) far faster than arithmetic ones and appear in compound interest, population growth, radioactive decay, and computer science.
Key Terms
| Term | Symbol | Meaning |
|---|---|---|
| First term | a | The starting value |
| Common ratio | r | The fixed multiplier between consecutive terms |
| nth term | an | The value of the term in position n |
Finding the Common Ratio
r = any term ÷ the term before it
Always check using at least two consecutive pairs to confirm the ratio is constant.
The nth Term Formula
an = a × rn−1
Where a = first term, r = common ratio, n = position of the term.
Sum of a Geometric Sequence
For r ≠ 1: Sn = a × (rn − 1) ÷ (r − 1) (when r > 1)
or equivalently: Sn = a × (1 − rn) ÷ (1 − r) (when r < 1)
Behaviour by Ratio Value
| r value | Behaviour | Example |
|---|---|---|
| r > 1 | Terms grow without limit | 2, 6, 18, 54 … (r=3) |
| 0 < r < 1 | Terms shrink towards zero | 100, 50, 25, 12.5 … (r=0.5) |
| r = 1 | All terms are equal | 5, 5, 5, 5 … |
| r < 0 | Terms alternate in sign | 3, −6, 12, −24 … (r=−2) |
Worked Examples
a = 5, r = 3. a8 = 5 × 37 = 5 × 2,187 = 10,935.
r = 48 ÷ 96 = 0.5. a5 = 96 × 0.54 = 96 × 0.0625 = 6.
a = 2, r = 3, n = 6. S6 = 2 × (36 − 1) ÷ (3 − 1) = 2 × (729 − 1) ÷ 2 = 728.
This is a geometric sequence: a = 1,000, r = 1.05.
a5 = 1,000 × 1.054 = 1,000 × 1.21551 = £1,215.51.
Key Takeaways
- Geometric sequence: constant ratio r between consecutive terms.
- nth term: an = a × rn−1.
- r > 1 → growth; 0 < r < 1 → decay; r < 0 → alternating signs.
- Compound interest, population growth, and radioactive decay all follow geometric sequences.
Practice Questions
- Find the 6th term of: 4, 12, 36, 108, …
- Find the common ratio and the 7th term of: 729, 243, 81, 27, …
- A sequence has first term 3 and common ratio 2. Find the sum of the first 8 terms.
- A city's population grows by 4% per year. Starting at 200,000, what is the population after 5 years?
- Which term of the sequence 2, 6, 18, … first exceeds 1,000?