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.
An old legend illustrates just how explosively geometric sequences grow: a wise man asks a king for a reward of one grain of rice on the first square of a chessboard, two on the second, four on the third, and so on, doubling each time across all 64 squares – a geometric sequence with a = 1 and r = 2. The king agrees, thinking it a modest request, but the total works out to 264 − 1 grains, roughly 18 quintillion – more rice than has ever been produced in all of human history. The same doubling effect drives Moore's Law, the semiconductor industry's observation, first made by Intel co-founder Gordon Moore in 1965, that the number of transistors on a computer chip roughly doubles every two years, a geometric sequence that shaped the entire pace of modern computing for decades.
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.