Chapter 2 · Sequences · Concept 2
2-8 Geometric Sequence (2)
You are told two terms and asked for the sequence. It worked in lesson 2-3 — write both facts, cancel the a, done. It works here too, with one surprise at the end: quite often there are two answers, and both are right.
Guiding question
Can two different sequences fit exactly the same facts?
Learning outcome. Calculate the general term of a geometric sequence, and determine certain terms in it.
Divide, do not subtract
The general term is still aₙ = a rⁿ⁻¹. Given two terms, write both, then divide one equation by the other — the a cancels and you are left with a power of r.
and the middle-number fact, matching 2b = a + c
b2 = ac
when a, b, c are geometric in that order
Why? b = ar and c = ar², so ac = a × ar² = a²r², which is (ar)² = b². In words: the middle number squared equals the outer two multiplied.
The step that decides everything is the power you are left with. Look at whether it is odd or even before you take the root.
Warm Up 1(a)
- press the button to start
Even power → TWO answers
r² = 4 → r = 2 or r = −2
Because (−2)² = 4 just as much as 2² = 4. A minus sign squared away leaves no trace, so both signs survive.
Give both sequences.
Odd power → ONE answer
r³ = −8 → r = −2 only
An odd power keeps the sign: 2³ = 8 and (−2)³ = −8. They are different numbers, so only one of them can equal −8.
One sequence, and that is the full answer.
The power you get is the gap between the two positions. From the 4th term to the 6th is a gap of 2 — even. From the 3rd to the 6th is a gap of 3 — odd. Count it before you divide.
1(a) The 4th term is 24 and the 6th term is 96. Find the general term.
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Say what you are using.
let aₙ = a rn−1Write this line every time — it costs nothing and markers look for it.
Work with a partner. Before either of you takes a root, both say out loud whether the power is odd or even, and how many answers you are expecting.
1. Find the general term.
A company plans a three-year budget in hundreds of thousands of pounds. Under a steady forecast, Year 1 is fixed at −5 (an infrastructure debt), and Years 2 and 3 are a and b; those three form an arithmetic sequence.
Under a breakthrough forecast, a, b and a projected Year 4 of 45 form a geometric sequence. Find a and b so that both plans hold at once.
Warm Up 1(a) gave two sequences: 3, 6, 12, 24, … and −3, 6, −12, 24, … . Both have 24 as their 4th term and 96 as their 6th. Is there any single extra fact the question could add that would leave only one of them?
Look at what the two sequences do differently.
Giving one answer when the power was even.
r² = 4 means r = 2 AND r = −2, and each has its own a. Two sequences, both worth writing down.
Reusing the same a for both branches.
With r = 2, a = 3. With r = −2, a = −3. Substitute each ratio back separately.
Subtracting the equations instead of dividing.
Subtracting leaves a(r⁵ − r³), which helps nobody. Dividing is what cancels the a here.
Getting the power wrong when dividing.
r⁵ ÷ r³ = r², not r⁸. Dividing powers means SUBTRACTING them.
Mixing up 2b = a + c and b² = ac.
The first is arithmetic, the second geometric. A question using both will tell you which order goes with which — read it twice.
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