Repeated multiplication by a constant ratio creates powerful patterns in sequences.
A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant ratio r.
u1, u2, u3, … is geometric
if un+1un = r for all n.
The nth term of a geometric sequence is given by
This formula allows us to find any term directly, without finding the previous terms.
A sequence is geometric if the ratio of consecutive terms is constant.
Example: 3, 9, 27, 81, …
93 = 3, 279 = 3, 8127 = 3
The ratio is constant (r = 3), so this is a geometric sequence.
Example: 2, 6, 12, 24, …
62 = 3, 126 = 2
(ratio not constant)
so this is not a geometric sequence.
The value of r determines the behaviour of the sequence.
Terms increase without bound.
(e.g. r = 2)
Terms increase without bound.
(e.g. r = 2)
Terms decrease towards 0.
(e.g. r = ½)
All terms are equal:
un = u1.
Terms change sign each step.
(e.g. r = −2)
Different values of r give different patterns of growth or decay.
Starting with un+1 = run,
we can derive un = u1rn−1.
u2 = ru1
u3 = ru2 = r(ru1) = r2u1
u4 = ru3 = r3u1
⋮
un = rn−1u1
Each step multiplies by r, so after n − 1 steps the term is u1rn−1.
Find the first five terms of the geometric sequence with u1 = 4 and r = 3.
EXPLORE THE SOLUTIONu₁ = 4
Find u10 for the geometric sequence with u1 = 5 and r = 2.
un = 5 × 2n−1
u10 = 5 × 29
= 5 × 512
= 2560
So the 10th term is 2560.
A later term can be used to find the ratio r, or the first term u1.
Example: Given u2 = 6 and u3 = 18, find r and u1.
r = u3u2 = 186 = 3.
u1 = u2r = 63 = 2.
So r = 3 and u1 = 2.
Geometric sequences appear in nature, finance and many real-world models where change is multiplicative.
Geometric sequences model situations where change is multiplicative.
A population that multiplies by a constant factor each period.
Compound interest grows by a constant percentage each period.
The value of an asset decreases by a constant percentage each year.
Arithmetic sequences add a constant difference. Geometric sequences multiply by a constant ratio.
| Property | Arithmetic | Geometric |
|---|---|---|
| Rule | un+1 = un + d | un+1 = run |
| Change | Constant difference d | Constant ratio r |
| Explicit | un = u1 + (n − 1)d | un = u1rn−1 |
| Example | 2, 5, 8, 11, … (d = 3) | 2, 6, 18, 54, … (r = 3) |
| Growth | Linear | Exponential |
A culture of bacteria starts with 500 cells and triples in number each hour.
Write down a recurrence relation for the number of cells after n hours.
[1]Find an explicit formula for un.
[2]Determine the least value of n for which un > 100 000.
[2]Interpret your answer in the context of the model.
[1]A geometric sequence is a discrete version of exponential growth: un = u1rn−1.
Solve problems such as un = k by using logarithms:
n = 1 + log(k/u1)log r
A = P(1 + i)n follows a geometric model.
Models of radioactive decay use un = u1(½)n.
Apply geometric sequences in finance, science and the environment.
“From a simple ratio comes a powerful pattern. Repeated multiplication turns numbers into exponential change.”
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ISSUE 01 / Geometric Sequences