EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Non-linear sequences
Algebra · Lesson 6 of 7
Teacher copy - includes the notes for whoever is teaching from it.
Last Lesson and Before
Answer each one, then check.
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
Learning Objectives
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)
Special Sequences
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Square numbers 1, 4, 9, 16, 25, ... - nth term n². |
Cube numbers 1, 8, 27, 64, 125, ... - nth term n³. |
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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. |
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Powers of 2 2, 4, 8, 16, 32, ... - nth term 2ⁿ. |
Geometric sequences Multiply by the same number each time: 3, 6, 12, 24, ... |
Triangular Numbers
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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. |
Each triangle adds a new row with one more dot than the last. |
Fibonacci-Type Sequences
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.
Geometric Sequences
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.
A Geometric Sequence
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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
PART TWO · HIGHER
Quadratic Sequences
When the differences themselves go up by the same amount.
Second Differences
2, 7, 14, 23, 34, ... The first differences change, but the second differences are all 2: the sequence is quadratic.
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n |
Term |
First difference |
Second difference |
|---|---|---|---|
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1 |
2 |
|
|
|
2 |
7 |
+5 |
|
|
3 |
14 |
+7 |
+2 |
|
4 |
23 |
+9 |
+2 |
|
5 |
34 |
+11 |
+2 |
The nth Term of a Quadratic Sequence
The nth term has the form an² + bn + c.
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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. |
The nth Term of a Quadratic Sequence
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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)
A Harder Quadratic Sequence
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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)
Geometric Sequences with Surds
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.
Case Study
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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. |
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1202 Liber Abaci is published |
1.618... The golden ratio, which the ratio of neighbouring terms approaches |
Key Terms
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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. |
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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. |
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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. |
Your Task: What Kind of Sequence?
12 minutes
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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. |
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.
Note: Sorting by type first stops students trying to find a common difference in a geometric sequence.
Can I...?
☐ 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.
Summary
✓ 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.
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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. |
Exam Practice: Non-Linear Sequences
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.
▸ 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.
Question 1 · 1 mark · Non-calculator
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“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.
Question 1 · mark scheme
1 mark available. Award a mark for each point made.
▸ 13. B1
▸ Model answer. 13
Question 2 · 2 marks · Non-calculator
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“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.
Question 2 · mark scheme
2 marks available. Award a mark for each point made.
▸ (a) 162. B1
▸ (b) 486. B1
▸ Model answer. (a) 54 × 3 = 162 (b) 162 × 3 = 486
Question 3 · 3 marks · Non-calculator · Higher
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“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.
Question 3 · mark scheme
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
▸ Model answer. The terms are a, b, a + b, a + 2b, 2a + 3b. So a + b = 7 and 2a + 3b = 18. Doubling the first: 2a + 2b = 14, so b = 4 and a = 3.
Question 4 · 3 marks · Non-calculator · Higher
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“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.
Question 4 · mark scheme
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
▸ Model answer. First differences 6, 10, 14, 18; second difference 4, so 2n². 2n² gives 2, 8, 18, 32, 50; the sequence minus 2n² is 1, 1, 1, 1, 1. nth term 2n² + 1.
Question 5 · 3 marks · Non-calculator · Higher
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“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.
Question 5 · mark scheme
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
▸ Model answer. First differences 5, 7, 9, 11; second difference 2, so n². n² gives 1, 4, 9, 16, 25; the sequence minus n² is −1, 1, 3, 5, 7, with nth term 2n − 3. nth term n² + 2n − 3.
Question 6 · 2 marks · Non-calculator · Higher
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“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.
Question 6 · mark scheme
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
▸ Model answer. 5th term = √3 × (√3)⁴ = √3 × 9 = 9√3