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Non-linear sequences - Teacher Slides.pptx
The lesson slides with the teacher's notes on each slide, and every question and mark scheme built in. Built from the lesson script on 28 September 2026.
EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Non-linear sequences
Algebra
Lesson 6 of 7
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Last Lesson and Before
Algebra
Lesson 6 of 7
Answer each one, then check.
1
Last lesson: find the nth term of 5, 8, 11, ...
2
Last lesson: work out the 10th term of 4n − 1.
3
Write down the first four square numbers.
4
Work out 2 × 2 × 2 × 2.
5
From the Number chapter: simplify √2 × √2.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Last Lesson and Before - Answers
Algebra
Lesson 6 of 7
1
Last lesson: find the nth term of 5, 8, 11, ...
3n + 2
2
Last lesson: work out the 10th term of 4n − 1.
39
3
Write down the first four square numbers.
1, 4, 9, 16
4
Work out 2 × 2 × 2 × 2.
16 = 2⁴
5
From the Number chapter: simplify √2 × √2.
2
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Learning Objectives
Algebra
Lesson 6 of 7
1
Recognise square, cube and triangular numbers and Fibonacci-type sequences.
2
Continue geometric sequences and find their common ratio.
3
Work out terms of a geometric sequence from its first term and ratio.
4
Find the nth term of a quadratic sequence. (Higher)
5
Work with geometric sequences with a surd as the ratio. (Higher)
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Special Sequences
Algebra
Lesson 6 of 7
Square numbers
1, 4, 9, 16, 25, ... - nth term n².
Cube numbers
1, 8, 27, 64, 125, ... - nth term n³.
Triangular numbers
1, 3, 6, 10, 15, ... - add 2, then 3, then 4, ...
Fibonacci sequence
1, 1, 2, 3, 5, 8, ... - each term is the sum of the two before it.
Powers of 2
2, 4, 8, 16, 32, ... - nth term 2ⁿ.
Geometric sequences
Multiply by the same number each time: 3, 6, 12, 24, ...
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Triangular Numbers
Algebra
Lesson 6 of 7
Each triangle adds a new row with one more dot than the last.
Triangular numbers count the dots in triangles like these. Each new triangle adds a row one dot longer than the last, so the differences go up by one each time: +2, +3, +4, ... That is what makes the sequence non-linear.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Fibonacci-Type Sequences
Algebra
Lesson 6 of 7
In a Fibonacci-type sequence, each term is the sum of the two terms before it.
Fibonacci
1, 1, 2, 3, 5, 8, 13, 21, ...
Any start
2, 5, 7, 12, 19, 31, ... follows the same rule from a different start.
With letters
Starting a, b, the terms are a, b, a + b, a + 2b, 2a + 3b, ...
Working back
If two neighbouring terms are 13 and 21, the one before is 21 − 13 = 8.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Geometric Sequences
Algebra
Lesson 6 of 7
A geometric sequence is multiplied by the same number, the common ratio r, every time.
Growing
3, 6, 12, 24, ... has r = 2.
Shrinking
64, 32, 16, 8, ... has r = ½.
Finding r
Divide any term by the one before it: 24 ÷ 12 = 2.
Any term
The nth term is (first term) × rⁿ⁻¹: the first term is multiplied by r once for each step after it.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
A Geometric Sequence
Algebra
Lesson 6 of 7
A geometric sequence starts 5, 15, 45, 135, ... Find the common ratio and the 8th term.
1
Divide a term by the one before
15 ÷ 5 = 3, so r = 3
2
The 8th term is 7 steps after the first
5 × 3⁷
3
Work out 3⁷
3⁷ = 2187
4
Multiply
5 × 2187 = 10 935
ANSWER
r = 3; 8th term = 10 935
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
PART TWO · HIGHER
Quadratic Sequences
When the differences themselves go up by the same amount.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Second Differences
Algebra
Lesson 6 of 7
2, 7, 14, 23, 34, ... The first differences change, but the second differences are all 2: the sequence is quadratic.
n
Term
First difference
Second difference
1
2
2
7
+5
3
14
+7
+2
4
23
+9
+2
5
34
+11
+2
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
The nth Term of a Quadratic Sequence
Algebra
Lesson 6 of 7
The nth term has the form an² + bn + c.
1
Second difference
Find it: it is always 2a.
2
The squared part
Halve the second difference to get a, and write an².
3
Subtract
Take an² away from each term.
4
The linear part
Find the nth term of what is left (bn + c).
5
Combine and check
an² + bn + c. Check with n = 1 and n = 2.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
The nth Term of a Quadratic Sequence
Algebra
Lesson 6 of 7
Find the nth term of 2, 7, 14, 23, 34, ...
1
Second difference 2, so a = 1
n²
2
n² gives
1, 4, 9, 16, 25
3
Sequence minus n²
1, 3, 5, 7, 9
4
The nth term of 1, 3, 5, 7, 9
2n − 1
5
Combine
n² + 2n − 1
ANSWER
n² + 2n − 1 (check n = 3: 9 + 6 − 1 = 14)
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
A Harder Quadratic Sequence
Algebra
Lesson 6 of 7
Find the nth term of 4, 13, 26, 43, 64, ...
1
First differences
9, 13, 17, 21
2
Second difference 4, so a = 2
2n²
3
2n² gives
2, 8, 18, 32, 50
4
Sequence minus 2n²
2, 5, 8, 11, 14, with nth term 3n − 1
5
Combine
2n² + 3n − 1
ANSWER
2n² + 3n − 1 (check n = 2: 8 + 6 − 1 = 13)
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Geometric Sequences with Surds
Algebra
Lesson 6 of 7
The common ratio can be a surd.
Example
√2, 2, 2√2, 4, 4√2, ... has r = √2, because √2 × √2 = 2.
Every other term
Two steps multiply by (√2)² = 2, so every second term doubles.
Finding a term
The 7th term of √3, 3, 3√3, ... is √3 × (√3)⁶ = √3 × 27 = 27√3.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Algebra
Lesson 6 of 7
CASE STUDY
Fibonacci and His Rabbits
In 1202 the Italian mathematician Leonardo of Pisa, later known as Fibonacci, published Liber Abaci, the book that helped bring Hindu-Arabic numerals to Europe. In it he set a puzzle about a pair of rabbits that breeds a new pair every month. The number of pairs month by month gives 1, 1, 2, 3, 5, 8, 13, ... - the sequence now named after him. Divide each term by the one before (8 ÷ 5 = 1.6, 13 ÷ 8 = 1.625, 21 ÷ 13 = 1.615…) and the answers close in on 1.618…, the golden ratio.
1202
Liber Abaci is published
1.618...
The golden ratio, which the ratio of neighbouring terms approaches
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Key Terms
Algebra
Lesson 6 of 7
Triangular numbers
1, 3, 6, 10, 15, ... - the numbers of dots in triangular patterns.
Fibonacci-type sequence
Each term is the sum of the two terms before it.
Geometric sequence
Each term is multiplied by the same number to get the next.
Common ratio
The number a geometric sequence is multiplied by each time.
Quadratic sequence (Higher)
A sequence whose second differences are all the same; its nth term includes n².
Second difference (Higher)
The difference between neighbouring first differences.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Algebra
Lesson 6 of 7
YOUR TASK
What Kind of Sequence?
12 minutes
For each sequence, say whether it is linear, geometric, Fibonacci-type or quadratic, and write down the next two terms. (a) 2, 6, 18, 54, ... (b) 3, 4, 7, 11, 18, ... (c) 4, 9, 14, 19, ... (d) 1, 4, 9, 16, ... (e) 80, 40, 20, 10, ... (f) 3, 6, 11, 18, 27, ... Higher: find the nth term of (f).
1
Check the differences.
2
Check the ratios.
3
Check whether each term is the sum of the two before.
WHAT A GOOD ANSWER SHOWS
(a) Geometric, r = 3: 162, 486. (b) Fibonacci-type: 29, 47. (c) Linear: 24, 29. (d) Quadratic (square numbers): 25, 36. (e) Geometric, r = ½: 5, 2.5. (f) Quadratic: 38, 51; nth term n² + 2.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Can I...?
Algebra
Lesson 6 of 7
Recognise square, cube and triangular numbers.
Continue a Fibonacci-type sequence.
Find the common ratio of a geometric sequence.
Work out a term of a geometric sequence.
Find second differences. (Higher)
Find the nth term of a quadratic sequence. (Higher)
Use a surd as a common ratio. (Higher)
Work backwards in a Fibonacci-type sequence.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Summary & Exam Focus
Special sequences: square, cube, triangular, Fibonacci, powers of 2.
Geometric: multiply by the common ratio r each time.
Fibonacci-type: add the two previous terms.
(Higher) Quadratic: constant second difference; a = second difference ÷ 2, then find the linear part of what is left.
EXAM FOCUS
Find an expression for the nth term of the quadratic sequence 3, 9, 19, 33, 51, ... (3 marks)
Write the first and second differences under the sequence before you do anything else. They tell you what kind of sequence it is and give you the first part of a quadratic nth term.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Exam Practice: Non-Linear Sequences
Algebra
Lesson 6 of 7
Answer all questions. Show your working. All questions are non-calculator. · 25 minutes
Question 1 · 1 mark · Non-calculator
Here are the first six terms of a Fibonacci sequence: 1, 1, 2, 3, 5, 8. Write down the next term.
Question 2 · 2 marks · Non-calculator
Here are the first four terms of a geometric sequence: 2, 6, 18, 54. (a) Write down the next term. (b) Work out the 6th term.
Question 3 · 3 marks · Non-calculator · Higher
The first two terms of a Fibonacci-type sequence are a and b. The 3rd term is 7 and the 5th term is 18. Find the values of a and b.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Exam Practice: Non-Linear Sequences (cont.)
Algebra
Lesson 6 of 7
Question 4 · 3 marks · Non-calculator · Higher
Find an expression for the nth term of the quadratic sequence 3, 9, 19, 33, 51, ...
Question 5 · 3 marks · Non-calculator · Higher
Find an expression for the nth term of the quadratic sequence 0, 5, 12, 21, 32, ...
Question 6 · 2 marks · Non-calculator · Higher
The first term of a geometric sequence is √3 and the common ratio is √3. Show that the 5th term is 9√3.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 1 · 1 mark · Non-calculator
Algebra
Lesson 6 of 7
“
Here are the first six terms of a Fibonacci sequence: 1, 1, 2, 3, 5, 8. Write down the next term.
HOW TO ANSWER IT
Command word: Non-calculator. Worth 1 mark, so plan before writing.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 1 · mark scheme
Algebra
Lesson 6 of 7
1 mark available. Award a mark for each point made.
13
B1
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 2 · 2 marks · Non-calculator
Algebra
Lesson 6 of 7
“
Here are the first four terms of a geometric sequence: 2, 6, 18, 54. (a) Write down the next term. (b) Work out the 6th term.
HOW TO ANSWER IT
Command word: Non-calculator. Worth 2 marks, so plan before writing.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 2 · mark scheme
Algebra
Lesson 6 of 7
2 marks available. Award a mark for each point made.
(a) 162
B1
(b) 486
B1
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 3 · 3 marks · Non-calculator · Higher
Algebra
Lesson 6 of 7
“
The first two terms of a Fibonacci-type sequence are a and b. The 3rd term is 7 and the 5th term is 18. Find the values of a and b.
HOW TO ANSWER IT
Command word: Non-calculator · Higher. Worth 3 marks, so plan before writing.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 3 · mark scheme
Algebra
Lesson 6 of 7
3 marks available. Award a mark for each point made.
Terms written in terms of a and b, at least to a + 2b
M1
a + b = 7 and 2a + 3b = 18
M1
a = 3 and b = 4
A1
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 4 · 3 marks · Non-calculator · Higher
Algebra
Lesson 6 of 7
“
Find an expression for the nth term of the quadratic sequence 3, 9, 19, 33, 51, ...
HOW TO ANSWER IT
Command word: Non-calculator · Higher. Worth 3 marks, so plan before writing.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 4 · mark scheme
Algebra
Lesson 6 of 7
3 marks available. Award a mark for each point made.
Second difference of 4, or 2n² seen
M1
Sequence minus 2n² found, e.g. 1, 1, 1, ...
M1
2n² + 1
A1
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 5 · 3 marks · Non-calculator · Higher
Algebra
Lesson 6 of 7
“
Find an expression for the nth term of the quadratic sequence 0, 5, 12, 21, 32, ...
HOW TO ANSWER IT
Command word: Non-calculator · Higher. Worth 3 marks, so plan before writing.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 5 · mark scheme
Algebra
Lesson 6 of 7
3 marks available. Award a mark for each point made.
Second difference of 2, or n² seen
M1
−1, 1, 3, 5, 7 or 2n − 3 found
M1
n² + 2n − 3
A1
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 6 · 2 marks · Non-calculator · Higher
Algebra
Lesson 6 of 7
“
The first term of a geometric sequence is √3 and the common ratio is √3. Show that the 5th term is 9√3.
HOW TO ANSWER IT
Command word: Non-calculator · Higher. Worth 2 marks, so plan before writing.
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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Question 6 · mark scheme
Algebra
Lesson 6 of 7
2 marks available. Award a mark for each point made.
√3 × (√3)⁴, or the terms √3, 3, 3√3, 9, 9√3 listed
M1
Correct working showing 9√3
A1
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