SHS2 Mathematics · Semester 1 Week 16: Patterns and Relations

Lesson notes

Learning Objectives

Indicator: 2.2.2.LI.2 - Recognise and find the nth term and the sum of the nth term of an arithmetic progression (AP) or linear sequence.

By the end of the lesson, learners can:

Recognise an arithmetic progression (AP) by verifying that consecutive terms have a constant difference, known as the common difference, d.
Derive and use the formula for the nth term of an AP, Un = a + d(n - 1), to find any specified term.
Derive and apply the formula for the sum of the first n terms of an AP, Sn = (n/2)(2a + (n - 1)d), to calculate partial sums.
Solve word problems involving APs set in everyday Ghanaian situations, interpreting the answers in context.
Distinguish an arithmetic (linear) sequence from other sequences by inspecting the pattern of differences between terms.

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Curriculum details

Strand: Algebraic Reasoning (Strand 2)
Sub-strand: Patterns and Relations (2.2)
Content standard: 2.2.2.CS.1 - Demonstrate understanding of patterns and relations involving sequence and series, generate strategies for algebraic formulas, and use them in solving real-life problems. 2.2.2.LO.1 Explore patterns of a sequence using plane figures and find the nth and the sum of the nth term of an arithmetic and geometric progression. Analyse, model, and solve real-life problems involving financial mathematics and exponential growth.
Indicator: 2.2.2.LI.2 - Recognise and find the nth term and the sum of the nth term of an arithmetic progression (AP) or linear sequence.
Suggested placement: Semester 1, Week 16 (Week 16 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.203: this exemplar sets a two-dimensional construct - a stacked fraction, an index or a column vector - which the text layer flattens into separate lines, so the parts are present but their vertical arrangement is lost; read the page
Curriculum reference: NaCCA curriculum document, p. 202

Exemplars (from the NaCCA curriculum)

Using Talk for Learning approaches, engage learners in a whole class to discuss the types of sequences as arithmetic (linear) and geometric (exponential) sequences. Encourage learners to volunteer to lead group discussions, leading to the development of leadership qualities for national development.
Using Collaborative Learning, create convenient groups and task learners to investigate, discuss and brainstorm the meaning of arithmetic progression as a sequence where the differences between every two terms are the same or every term is obtained by adding a fixed number (positive or negative or zero) to its previous term. (e.g., 3, 6, 9, 12, 15, ....) and establish the general rule for the nth term as 𝑈𝑛 = 𝑎 + 𝑑(𝑛 − 1) where a is the first term, d is the common difference, and n is the number of terms and the sum of the first n terms as 𝑆 𝑛 = (2𝑎 + (𝑛 − 1)𝑑 by creating different numerical 𝑛 2 sequences and investigating the patterns.
Example 1: Find the formula for the nth term of the arithmetic sequence 2, 5, 8, ... and find the 6 th term.
Solution 𝑎 = 2 and 𝑑 = 3 So 𝑈 𝑛 = 2 + 3(𝑛 − 1) ∴ 𝑈 𝑛 = 3𝑛 − 1 𝑛 = 6 ∴ 𝑈 𝑛 = 3(6) − 1 = 17 Therefore, the 6 th term of the above sequence is 17.
Example 2: Find the sum of the first 10 terms of the arithmetic sequence 3, 7, 11, ...
Solution 𝑛 𝑆 𝑛 = (2𝑎 + (𝑛 − 1)𝑑 2 𝑛 = 10, 𝑎 = 3 𝑎𝑛𝑑 𝑑 = 4 10 𝑆 10 = (2 × 3 + (10 − 1) × 4 2 𝑆 10 = 210
Teaching and Learning Resources:
- * Matchsticks * Cardboard
Assessment (2.2.2.AS.2). The document marks these depth-of-knowledge levels for this indicator: Level 1 Recall; Level 2 Skills of conceptual understanding; Level 3 Strategic reasoning.