Cambridge Lower Secondary Mathematics Stages 7 | Hoaii
UNIT 9 – SEQUENCES AND FUNCTIONS
THE GEOMETRIC SEQUENCES
1. What is a geometric sequence?
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A geometric sequence is an ordered set of numbers that progresses by multiplying or
dividing each term by a common ratio (r).
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A geometric sequence can be known as a geometric progression
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If we multiply or divide by the same number each time to make the sequence, it is a
geometric sequence.
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Some examples of geometric sequences
First term
Term-to-term rule
First 5 terms
3
Multiply by 3
3, 9, 27, 81, 243, …
5
Multiply by 2
5, 10, 20, 40, 80, …
-12
Divide by 2
– 12, – 6, – 3, – 1.5, – 0.75, …
0.8
Multiply by 5
0.8 ; 4 ; 20 ; 100 ; 500 ; …
2. Geometric sequence formula (The nth term formula)
an = a1 x (r)n-1
 an : the nth term (general term)
 a1 : the first term
 n : the term position
 r : the common ratio
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Cambridge Lower Secondary Mathematics Stages 7 | Hoaii
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For example: Find the 50th in the geometric sequence below
5 ; 10 ; 20 ; 40 ; 80 ; …
 The common ratio is 2 so r = 2
 a1 = 5
 n = 50
 a50 = 5 x 250-1 = 5 x 249
3. Type 1: How to continue a geometric sequence
 Step 1: Take two consecutive terms from the sequence.
 Step 2: Divide the second term by the first term to find the common ratio (r)
 Step 3: Multiply the last term in the sequence by the common ratio to find the next
term. Repeat for each new term.
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For example: Calculate the next 3 terms of the sequence – 1 ; – 3 ; – 9 ; – 27 ; – 81 ; …
 Choose two consecutive terms. There are – 27 and – 9
 Common ratio: - 27 ÷ - 9 = 3
 - 81 x 3 = - 243
- 243 x 3 = - 729
- 729 x 3 = - 2187
4. Type 2: How to find missing numbers in a geometric sequence
 Step 1: Calculate the common ratio between two consecutive terms.
 Step 2: Multiply the term before any missing value by the common ratio or divide the
term after any missing value by the common ratio.
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For example: Fill in the missing terms in the sequence - 4 ; … ; … ; - 108 ; …
 First we need to find the factor between the two terms: - 108 ÷ - 4 = 27. But to get
from – 108 to - 4, we jump 3 terms. This means that – 4 has been multiplied by the
common ratio three times or – 4 x r x r x r = - 4r3
 r=3
 - 4 x 3 = - 12
- 12 x 3 = - 36
- 108 x 3 = - 324
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