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What Is a Sequence? Patterns, Terms, and Notation
Introduction to Sequences · Discovery/Inquiry
This unit opens the chapter at the concrete level of familiar number patterns, using dot arrays to anchor the abstract idea of a term's position before notation is formalised. A common confusion at this stage is treating the position index n and the term value tn as interchangeable — watch for students who write t3 = 3 for the triangular-number sequence rather than t3 = 6. Students who finish early may explore Exercise Set 8.1 Q1, p.179 (E-10) as a self-study extension.
0-12Display four sequences on the board — natural numbers, odd numbers, triangular numbers, square numbers — and ask students in pairs to describe the pattern in each and predict the next three terms. Collect predictions and discuss: what makes each list a sequence? Establish the definition of a sequence as an ordered list of numbers (terms), and distinguish finite from infinite sequences.
12-28Use the dot-array figures from the textbook to show how triangular numbers arise as running sums of natural numbers (1, 1+2, 1+2+3, ...) and square numbers as running sums of odd numbers. Introduce the subscript notation t1, t2, t3, ... and note that the position index n is always a positive integer while term values may be any real number. Pose the think-and-reflect question: list five different types of sequences and discuss one property each has — students write briefly, then share.
28-42Students work individually on Exercise, p.176 (E-1): predict the next four terms of 1, 4, 7, 10, 13, ... and build the running-sum sequence. Then tackle Exercise, p.176 (E-2): write t5 through t8 for the triangular-number sequence using the dot-array pattern.
42-50Whole-class debrief: invite students to share their running-sum sequences and triangular-number terms. Clarify the difference between a term's value and its position, and ensure every student can read and write the subscript notation correctly.
Competency (C-9.2). Read and analyse a sequence as a rule that generates terms one after another, distinguishing explicit from recursive rules.
Learning outcome. Students can describe the pattern in familiar number sequences (natural, odd, triangular, square) and predict further terms using the subscript notation tn.
The triangular-number sequence begins 1, 3, 6, 10, 15, ... The 7th triangular number t7 is:
- A 21
- B 25
- C 28
- D 36
C. 28
What each wrong option reveals
- A Adds 6 to t6 = 15 (correct step) but then stops at 21 — this is actually t6 computed correctly as t6 = 1+2+3+4+5+6 = 21, not t7; student may have miscounted the term position.
- B Applies a square-number formula (5^2 = 25 or 7+18 from a mistaken pattern) rather than summing consecutive integers to build the triangular number.
- D Confuses the 7th triangular number with the 7th square number (7^2 = 49 — or picks t8 = 36 by a one-off error in the term index).
Also saved as 7 periods or 10 periods.
- 2Computing Any Term Directly: The Explicit Rule
Explicit Rule for a Sequence - 3Building Each Term from the Last: The Recursive Rule
Recursive Rule for a Sequence - 4Constant Differences: Defining the Arithmetic Progression
Arithmetic Progressions - 5Plotting an AP: Straight-Line Patterns and the Recursive Form
Visualising an AP - 6Summing Natural Numbers: Deriving the Formula and Its Applications
Sum of the First n Natural Numbers - 7Constant Ratios: Defining the Geometric Progression
Geometric Progressions - 8Finding Any GP Term: Explicit and Recursive Formulas in Practice
Geometric Progressions - 9Fractals and the Sierpinski Triangle: GPs in Self-Similar Patterns
Fun with Fractals - 10Plotting a GP: Curves, Bouncing Balls, and Real Contexts
Visualising a GP - 11Consolidating Progressions: AP and GP Problems from Exercise Set 8.2 and 8.3
Arithmetic Progressions (Revisit) - 12Chapter Synthesis: Connecting Sequences, APs, GPs, and Their Rules
Synthesis
- 1. Orienting Yourself: The Use of Coordinates
- 2. Introduction to Linear Polynomials
- 3. The World of Numbers
- 4. Exploring Algebraic Identities
- 5. I'm Up and Down, and Round and Round
- 6. Measuring Space: Perimeter and Area
- 7. The Mathematics of Maybe: Introduction to Probability
- 8. Predicting What Comes Next: Exploring Sequences and Progressions
- 9. Propositions and their Converses
- 10. How Quantities Combine: Understanding Data
- 11. The World of Algorithms
- 12. Quadrilaterals
- 13. Two Variables, One Line
- 14. Math of Space: Surface Area and Volume
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