Chapter 2 · Sequences · Concept 2
2-7 Geometric Sequence (1)
One E. coli cell enters the blood. It splits in two every twenty minutes, and each of those splits again. Nothing is being added here — everything is being multiplied, and after a single day that one cell has become more bacteria than there are stars in the galaxy.
Guiding question
What happens to a sequence when you multiply instead of add?
Learning outcome. Calculate the general term of a geometric sequence, and determine certain terms in it.
Geometric sequences
If you get each term by multiplying the one before by the same number, the sequence is geometric, and that number is the common ratio, written r.
1, ×2 2, ×2 4, ×2 8, ×2 16, …
first term 1, common ratio 2
the general term, with first term a and ratio r
aₙ = a rⁿ⁻¹
To find r, divide any term by the one before it. Do it twice to check — if the two answers disagree, the sequence is not geometric.
| Arithmetic (2-2) | Geometric (here) | |
|---|---|---|
| how you move on | add d | multiply by r |
| how you find it | subtract two terms | divide two terms |
| the general term | a + (n − 1)d | a rⁿ⁻¹ |
| why n − 1 | the first term is free | the first term is free |
The (n − 1) is here for the same reason.
You do not multiply to get to term 1 — you are already there. So term 5 has been multiplied four times, which is why the power is n − 1. Test it: at n = 1 the power is 0, and r⁰ = 1, so a₁ = a ✓
Arithmetic sequences only ever go up or down in a straight line. Geometric ones do three quite different things depending on r — and you need to recognise all three on sight.
doubles every step
aₙ = 2 × (2)n−1
r bigger than 1
runs away, faster and faster
r between 0 and 1
shrinks towards zero, never quite reaching
r negative
flips sign every single step
1(c) Find the general term of −4, 6, −9, 27/2, …
- 1
Write down the first term. It is just the first number.
a = −4No working needed. Do not lose the minus.
Work with a partner. Whoever finds r first must check it on a second pair before the other one is allowed to write the formula.
1. Find the general term.
2. Find the ratio and the missing letters.
In an increasing geometric sequence, a₃ + a₅ = 80 and a₄ = 24. Find a₇.
A bacterium doubles every 20 minutes. An arithmetic sequence starting at 1 and adding 1,000,000 every 20 minutes sounds far more impressive at the start. Which one is ahead after 24 hours, and what does that tell you about the two kinds of growth?
24 hours is 72 doublings. You do not need the exact number — just compare the shapes.
Writing rⁿ instead of rⁿ⁻¹.
Test at n = 1: the formula must give back the first term. r⁰ = 1 makes that work; rⁿ gives you a × r, which is the second term.
Dropping the brackets round a negative or fractional ratio.
2 × (−3)ⁿ⁻¹ is right. 2 × −3ⁿ⁻¹ means something else entirely — the power lands only on the 3.
Subtracting to find r.
That is the arithmetic move. Here you divide: r = a₂ ÷ a₁.
Multiplying when you should divide going backwards.
In A, 24, 12, … the A comes BEFORE, so A = 24 ÷ r, not 24 × r.
Checking r on only one pair.
Any two numbers have a ratio. It takes a second pair agreeing to prove the sequence is geometric at all.
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