SHS2 Additional Mathematics · Semester 1 Week 4: Application of Algebra

Lesson notes

Learning Objectives

Indicator: 2.1.1.LI.2 - Use recursive and explicit formulae of sequences to model situations and translate between the two forms.

By the end of the lesson, learners can:

Identify whether a given sequence is defined recursively or explicitly and generate terms from either form.
Write a recursive formula with its initial condition(s) for a given sequence, including arithmetic and geometric sequences and more complex recurrence relations such as aₙ = 3aₙ₋₁ + 2.
Derive an explicit formula for the nth term of a sequence from its recursive definition, and vice versa, translating between the two forms.
Model a real-life situation (such as savings, population growth, or loan repayment) using both recursive and explicit formulae, and use either form to find specific terms.
Use the explicit formula to find any term directly without generating all preceding terms, and explain when each form is more convenient.

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Curriculum details

Strand: Modelling with Algebra (Strand 1)
Sub-strand: Application of Algebra (1.1)
Content standard: 2.1.1.CS.2 - Demonstrate the ability to apply algebraic processes and reasoning to model and solve real life situations involving sequences, and linear programming and use appropriate techniques to solve quadratic inequalities, as well as resolve rational functions. 2.1.1.LO.1 Investigate De Morgan's law on sets algebraically and graphically, formulate and solve real life problems up to three sets. 2.1.1.LO.2 Model sequence recursively and explicitly, and establish the relationship between the two forms, as well as solve real life problems involving linear and exponential sequences and series. 2.1.1.LO.3 Apply indices and logarithms to solve real life problems, including logarithms with different bases, and sketch and interpret logarithmic functions. 2.1.1.LO.4 Formulate and derive appropriate strategies to solve quadratic inequalities. 2.1.1.LO.5 Graph systems of given inequality and identify the region that provides the feasible solution and apply it to real life situations. 2.1.1.LO.6 Determine the set of values for which a rational function is defined and resolve rational functions into partial fractions. 2.1.1.LO.7 Multiply matrices, determine the inverse of a 2 x 2 matrix, find the determinant up to a 3 x 3 matrix and represent matrices in linear transformations.
Indicator: 2.1.1.LI.2 - Use recursive and explicit formulae of sequences to model situations and translate between the two forms.
Suggested placement: Semester 1, Week 4 (Week 4 of the year)
Our suggestion, laid out in curriculum order across three terms of twelve weeks. NaCCA does not fix the week, so follow your school's scheme of learning.
Source document note: The official curriculum document has a fault in this entry, so the curriculum text above is transcribed exactly as printed:

exemplars - p.286: adjacent duplicated CambriaMath characters were collapsed, but this PDF also maps some equation glyphs to the wrong letter or operator; verify every mathematical expression in this row against the rendered page
Curriculum reference: NaCCA curriculum document, p. 286

Exemplars (from the NaCCA curriculum)

Collaborative Learning: Talk for Learning Approaches, Project-based Learning
Experiential Learning and Problem-based Learning
Activity: Recursive sequences
Use Talk for Learning Approaches, Problem-based Approaches, Experiential and Collaborative Learning Approaches to solve problems related to recursive sequences. Learners recognise recursive sequences, work in their groups to investigate recurrence sequences and use initial conditions to generate terms of the sequence as well as iterative rules.
Example Write the recurring decimals as a series. Identify the common ratio and the first and an explicit formula for the nth term.
- 0.3333...
- 0.54545454...
- 3.212121...
Solution
- 0.3333... = 0.3 + 0.3(0.1) + 0.3(0.1) 2 + 0.3(0.1) 3 + .. The first term is 0.3, and the constant ratio between each successive term is 0.1. The formula for the nth term will be 𝑈 % = 0.3( !( ) %*! !
Find the recurrence relation and the initial conditions for the sequence: 1, 5, 17, 53, 161, 485....
Solution: The recurrence formula is 𝑎 % = 3𝑎 %*! + 2, and the initial condition is 𝑎 ( = 1 A recursive rule for an AP is 𝑎 ! = −3; 𝑎 % = 𝑎 %*! + 7. What is the iterative rule for this sequence?
- 𝑎 % = 7𝑛 − 10
- 𝑎 % = −3𝑛 + 7
Find the next two terms in (𝑎 % ), 𝑛 ≥ 0 for the sequence 3, 5, 11, 21, 43, 85...... the recurrence relation and initial conditions for the sequence
Teaching and Learning Resources:
- Worksheets
- Scientific Calculator
- Technological tools, apps, etc.
- SHS Additional Mathematics Curriculum
Assessment (2.1.1.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.