Patterns and Relations · SHS2 Mathematics

Lesson notes

Learning Objectives

Indicator: 2.2.2.LI.3 - Identify geometric progression or exponential sequence and find the algebraic expression for the general term.

By the end of the lesson, learners can:

Identify whether a given sequence is a geometric progression (GP) or exponential sequence by checking for a constant ratio between consecutive terms.
Define the common ratio, r, of a geometric progression and state the general term formula Uₙ = arⁿ⁻¹, where a is the first term, r is the common ratio, and n is the term number.
Find the nth term of a geometric progression given the first term, common ratio, and term number.
Determine the first term and common ratio of a GP when two non-consecutive terms are given, using simultaneous equations.
Use the general term formula to solve practical problems involving geometric sequences in Ghanaian contexts.

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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.3 - Identify geometric progression or exponential sequence and find the algebraic expression for the general term.
Suggested placement: Semester 1, Week 17 (Week 17 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.204: 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. 203

Exemplars (from the NaCCA curriculum)

Collaborative Learning: In small ability groups, task learners to investigate, discuss and brainstorm the meaning of geometric sequence (exponential sequence) as a sequence in which each term is obtained by multiplying the previous term by a constant factor. ( e.g., 6, 12, 24, 48...), and establish the general rule for the nth term as 𝑈 𝑛 = 𝑎𝑟 𝑛−1 where 𝑎 is the first term, 𝑟 is the common ratio 𝑛 is the number of terms, and the sum 𝑎(𝑟 𝑛 −1) 𝑎(1−𝑟 𝑛 ) of the first n terms as 𝑆 𝑛 = where 𝑟 > 1 and 𝑆 𝑛 = where 𝑟 < 1, by creating different 𝑟−1 1−𝑟 numerical sequences and investigating the patterns.
Example 1: Find the 8th term of the exponential sequence 2, 6, 18, 54...
Solution 𝑈 𝑛 = 𝑎𝑟 𝑛−1 𝑎 = 2, 𝑟 = 3, 𝑛 = 8 𝑈 8 = 2 × 3 8−1 𝑈 8 = 4372
1 Example 2: Find the sum of the first 7 terms of the G P , 1, 2, 4, ... 2
Solution 𝑎(𝑟 𝑛 −1) 1 𝑆 𝑛 = since 𝑟 > 1 𝑟 = 2, 𝑎 = , 𝑛 = 7 𝑟−1 2 1 7 (2 −1) 𝑆 7 = 2 2−1 1 𝑆 7 = (256 − 1) 2 𝑆 7 = 127.5
Example 3: The second and the fourth terms of an exponential sequence (GP) are 9 and 4, respectively. Find the sequence.
Solution 𝑈 2 = 𝑎𝑎 = 9 ... ... .(1) 𝑈 4 = 𝑎𝑟 3 = 4 ... ... . (2) 2 Solving the equation simultaneously 𝑟 = 3 and 𝑎 = 13.5 Using 𝑎, 𝑎𝑎, 𝑎𝑟 2 , 𝑎𝑟 3 ... the sequence is 13.5, 9, 6, ...
Teaching and Learning Resources:
- * Matchsticks * Cardboard
Assessment (2.2.2.AS.3). The document marks no depth-of-knowledge level for this indicator.