GRADE 12 PURE MATHEMATICS | BOOK 1
GRADE 12 PURE MATHEMATICS
BOOK 1
Patterns, Sequences, Series
and Financial Mathematics
LEARN • UNDERSTAND • PRACTISE • APPLY
Grade 12 | South African CAPS/NSC study resource
Reviewed working edition
Book 1 • Reviewed working edition | Page 1
, GRADE 12 PURE MATHEMATICS | BOOK 1
CONTENTS
Book 1 is organised into three learning blocks. Each chapter includes explanations, worked examples, practice,
examination-level questions and memoranda.
Chapter 1: Patterns and Sequences
Chapter 2: Arithmetic Sequences
Chapter 3: Geometric Sequences
Chapter 4: Arithmetic Series
Chapter 5: Geometric Series
Chapter 6: Sigma Notation
Chapter 7: Sum to Infinity of a Geometric Series
Chapter 8: Quadratic Patterns and Mixed Sequence Problems
Chapter 9: Foundations of Financial Mathematics
Chapter 10: Growth, Decay and Depreciation
Chapter 11: Future-Value Annuities
Chapter 12: Present-Value Annuities and Loans
Chapter 13: Time Periods Using Logarithms
Chapter 14: Investment and Loan Decision-Making
Chapter 15: Integrated Financial Mathematics Problems
Mathematical notation and financial examples are presented for study and practice. Read each question
carefully, show working, and interpret answers in context.
Book 1 • Reviewed working edition | Page 2
, GRADE 12 PURE MATHEMATICS | BOOK 1
CHAPTER 1
Patterns and Sequences
1. START HERE
Mathematical patterns appear in number lists, diagrams, arrangements and real-life situations. A sequence is
an ordered list of terms generated by a rule. Your first job is to identify how the terms change; only then
should you choose a formula.
For example, 4; 7; 10; 13; … increases by 3 each time. The pattern is predictable, so the next term is 16. Other
sequences may multiply by a fixed number or have differences that change in a regular way.
2. Learning Objectives
• recognise and describe a numerical pattern;
• identify terms and their positions in a sequence;
• calculate first and second differences;
• distinguish arithmetic, geometric and quadratic patterns at an introductory level; and
• explain why a proposed term fits or does not fit a rule.
3. Concept Explanation from First Principles
3.1 Look at how terms change
Subtract consecutive terms to find first differences. If these differences are constant, the sequence is
arithmetic. For example, 2; 6; 10; 14; … has first differences 4; 4; 4.
3.2 Look for a constant ratio
Divide each term by the term before it, where the division is defined. If the ratio is constant, the sequence is
geometric. For example, 3; 6; 12; 24; … has common ratio 2.
3.3 Look at second differences
If first differences are not constant, calculate differences between those differences. Constant second
differences indicate a quadratic pattern. For example, 1; 4; 9; 16; 25; … has first differences 3; 5; 7; 9 and
second differences 2; 2; 2.
3.4 Describe the rule clearly
A rule may describe how to get the next term, or it may give a formula for the term at position n. In this
introductory chapter, focus on recognising and explaining the pattern. The next chapters develop the formulae
in detail.
4. Key Definitions
Pattern: A rule or regularity connecting objects or numbers.
Sequence: An ordered list of terms following a rule.
Term: A number or object in a sequence.
Position: The place occupied by a term, often labelled n.
First difference: The difference between consecutive terms.
Book 1 • Reviewed working edition | Page 3
, GRADE 12 PURE MATHEMATICS | BOOK 1
Second difference: The difference between consecutive first differences.
Common ratio: The quotient of a term divided by the preceding term, when defined.
5. Principles to Remember
• Constant first differences → arithmetic pattern.
• Constant ratio → geometric pattern.
• Constant second differences → quadratic pattern.
• Write the terms in order and show the differences clearly.
• A suggested rule should be checked against more than one pair of terms.
6. Detailed Worked Examples
Worked Example 1 — Continue a pattern
Find the next two terms: 5; 9; 13; 17; …
First differences: 4; 4; 4. Add 4 each time.
Next terms: 21 and 25.
Worked Example 2 — Identify a geometric pattern
Describe 2; 6; 18; 54; … and find the next term.
Ratios: 6/2 = 3, 18/6 = 3, 54/18 = 3.
The common ratio is 3, so the next term is 54 × 3 = 162.
Worked Example 3 — Use second differences
Examine 2; 6; 12; 20; 30; …
First differences are 4; 6; 8; 10. Second differences are 2; 2; 2. The sequence has a quadratic pattern.
The next first difference is 12, so the next term is 30 + 12 = 42.
Worked Example 4 — Explain why a sequence is not arithmetic
Is 3; 6; 12; 24; … arithmetic?
First differences are 3; 6; 12, which are not constant. Therefore, it is not arithmetic. Its ratio is constantly 2, so
it is geometric.
7. LOOK FOR — Recognising What a Question Asks
• “Find the next term” → identify and continue the rule.
• “Classify the sequence” → test first differences, ratios and then second differences.
• “Explain your answer” → show the calculation that reveals the pattern.
• “Does this number occur?” → use the rule to test whether the value can appear at a valid position.
Book 1 • Reviewed working edition | Page 4
GRADE 12 PURE MATHEMATICS
BOOK 1
Patterns, Sequences, Series
and Financial Mathematics
LEARN • UNDERSTAND • PRACTISE • APPLY
Grade 12 | South African CAPS/NSC study resource
Reviewed working edition
Book 1 • Reviewed working edition | Page 1
, GRADE 12 PURE MATHEMATICS | BOOK 1
CONTENTS
Book 1 is organised into three learning blocks. Each chapter includes explanations, worked examples, practice,
examination-level questions and memoranda.
Chapter 1: Patterns and Sequences
Chapter 2: Arithmetic Sequences
Chapter 3: Geometric Sequences
Chapter 4: Arithmetic Series
Chapter 5: Geometric Series
Chapter 6: Sigma Notation
Chapter 7: Sum to Infinity of a Geometric Series
Chapter 8: Quadratic Patterns and Mixed Sequence Problems
Chapter 9: Foundations of Financial Mathematics
Chapter 10: Growth, Decay and Depreciation
Chapter 11: Future-Value Annuities
Chapter 12: Present-Value Annuities and Loans
Chapter 13: Time Periods Using Logarithms
Chapter 14: Investment and Loan Decision-Making
Chapter 15: Integrated Financial Mathematics Problems
Mathematical notation and financial examples are presented for study and practice. Read each question
carefully, show working, and interpret answers in context.
Book 1 • Reviewed working edition | Page 2
, GRADE 12 PURE MATHEMATICS | BOOK 1
CHAPTER 1
Patterns and Sequences
1. START HERE
Mathematical patterns appear in number lists, diagrams, arrangements and real-life situations. A sequence is
an ordered list of terms generated by a rule. Your first job is to identify how the terms change; only then
should you choose a formula.
For example, 4; 7; 10; 13; … increases by 3 each time. The pattern is predictable, so the next term is 16. Other
sequences may multiply by a fixed number or have differences that change in a regular way.
2. Learning Objectives
• recognise and describe a numerical pattern;
• identify terms and their positions in a sequence;
• calculate first and second differences;
• distinguish arithmetic, geometric and quadratic patterns at an introductory level; and
• explain why a proposed term fits or does not fit a rule.
3. Concept Explanation from First Principles
3.1 Look at how terms change
Subtract consecutive terms to find first differences. If these differences are constant, the sequence is
arithmetic. For example, 2; 6; 10; 14; … has first differences 4; 4; 4.
3.2 Look for a constant ratio
Divide each term by the term before it, where the division is defined. If the ratio is constant, the sequence is
geometric. For example, 3; 6; 12; 24; … has common ratio 2.
3.3 Look at second differences
If first differences are not constant, calculate differences between those differences. Constant second
differences indicate a quadratic pattern. For example, 1; 4; 9; 16; 25; … has first differences 3; 5; 7; 9 and
second differences 2; 2; 2.
3.4 Describe the rule clearly
A rule may describe how to get the next term, or it may give a formula for the term at position n. In this
introductory chapter, focus on recognising and explaining the pattern. The next chapters develop the formulae
in detail.
4. Key Definitions
Pattern: A rule or regularity connecting objects or numbers.
Sequence: An ordered list of terms following a rule.
Term: A number or object in a sequence.
Position: The place occupied by a term, often labelled n.
First difference: The difference between consecutive terms.
Book 1 • Reviewed working edition | Page 3
, GRADE 12 PURE MATHEMATICS | BOOK 1
Second difference: The difference between consecutive first differences.
Common ratio: The quotient of a term divided by the preceding term, when defined.
5. Principles to Remember
• Constant first differences → arithmetic pattern.
• Constant ratio → geometric pattern.
• Constant second differences → quadratic pattern.
• Write the terms in order and show the differences clearly.
• A suggested rule should be checked against more than one pair of terms.
6. Detailed Worked Examples
Worked Example 1 — Continue a pattern
Find the next two terms: 5; 9; 13; 17; …
First differences: 4; 4; 4. Add 4 each time.
Next terms: 21 and 25.
Worked Example 2 — Identify a geometric pattern
Describe 2; 6; 18; 54; … and find the next term.
Ratios: 6/2 = 3, 18/6 = 3, 54/18 = 3.
The common ratio is 3, so the next term is 54 × 3 = 162.
Worked Example 3 — Use second differences
Examine 2; 6; 12; 20; 30; …
First differences are 4; 6; 8; 10. Second differences are 2; 2; 2. The sequence has a quadratic pattern.
The next first difference is 12, so the next term is 30 + 12 = 42.
Worked Example 4 — Explain why a sequence is not arithmetic
Is 3; 6; 12; 24; … arithmetic?
First differences are 3; 6; 12, which are not constant. Therefore, it is not arithmetic. Its ratio is constantly 2, so
it is geometric.
7. LOOK FOR — Recognising What a Question Asks
• “Find the next term” → identify and continue the rule.
• “Classify the sequence” → test first differences, ratios and then second differences.
• “Explain your answer” → show the calculation that reveals the pattern.
• “Does this number occur?” → use the rule to test whether the value can appear at a valid position.
Book 1 • Reviewed working edition | Page 4
