Lesson Notes By Weeks and Term v4 - SHS 2
APPLICATION OF ALGEBRA
Download the Lessonotes Mobile Ghana app for faster lesson access on Android and iPhone.
APPLICATION OF ALGEBRA
Download the Lessonotes Mobile Ghana app for faster lesson access on Android and iPhone.
Subject: Additional Mathematics
Class: SHS 2
Term: 1st Term
Week: 4
Grade code: 2.1.1.LI.2
Strand code: 1
Sub-strand code: 1
Content standard code: 2.1.1.CS.2
Indicator code: 2.1.1.LI.2
Theme: MODELLING WITH ALGEBRA
Subtheme: APPLICATION OF ALGEBRA
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
This lesson explores two powerful ways to describe patterns in numbers, known as sequences. We find sequences everywhere in Ghana – from the beautiful repeating patterns in Kente designs to how our savings grow with Mobile Money interest, or even how a plant grows taller each week. We will learn how to describe these patterns using explicit formulae (a direct rule to find any term) and recursive formulae (a step-by-step rule to get from one term to the next). Understanding both methods allows us to model and solve real-world problems more effectively.
A. What is a Sequence? A sequence is an ordered list of numbers, called terms. We use notation like `U_n` or `a_n` to represent the term in the *n*th position. Example: In the sequence 2, 4, 6, 8, ... The first term is `a_1 = 2`. The second term is `a_2 = 4`. The *n*th term is `a_n`.
There are two main ways to write a rule, or formula, for a sequence. B. The Explicit Formula An explicit formula gives you a direct way to calculate *any* term in the sequence just by knowing its position, `n`. Think of it as a "direct flight" to the term you want. Key Idea: It defines `a_n` in terms of `n`. Example 1 (Arithmetic Progression): Consider the sequence 5, 8, 11, 14, ... The first term is `a_1 = 5`. The common difference is `d = 3`. The explicit formula is `a_n = a_1 + (n-1)d`. Substituting our values: `a_n = 5 + (n-1)3 = 5 + 3n - 3 = 3n + 2`. To find the 10th term, we just substitute `n=10`: `a_10 = 3(10) + 2 = 32`. Example 2 (Geometric Progression): Consider the sequence 2, 6, 18, 54, ... The first term is `a_1 = 2`. The common ratio is `r = 3`. The explicit formula is `a_n = a_1 * r^(n-1)`. Substituting our values: `a_n = 2 * 3^(n-1)`. To find the 5th term: `a_5 = 2 * 3^(5-1) = 2 * 3^4 = 2 * 81 = 162`. C. The Recursive Formula (or Recurrence Relation) A recursive formula defines a term based on the term(s) that came *before* it. Think of it as giving someone directions step-by-step. For this to work, you MUST provide a starting point, called the initial condition. Key Idea: It defines `a_n` in terms of `a_(n-1)` (the previous term). It has two parts: The initial condition(s) (e.g., `a_1 = ...`). The recursive rule (the formula connecting a term to the previous one). Example 1 (Arithmetic Progression): The sequence 5, 8, 11, 14, ... Initial Condition: `a_1 = 5`. Recursive Rule: To get any new term, you add 3 to the previous term. So, `a_n = a_(n-1) + 3`. Example 2 (Geometric Progression): The sequence 2, 6, 18, 54, ... Initial Condition: `a_1 = 2`. Recursive Rule: To get any new term, you multiply the previous term by 3. So, `a_n = 3 * a_(n-1)`. Example 3 (More Complex Relation): Consider the rule: `a_1 = 1` and `a_n = 3a_(n-1) + 2`. `a_1 = 1` (This is given). `a_2 = 3 * a_1 + 2 = 3(1) + 2 = 5`. `a_3 = 3 * a_2 + 2 = 3(5) + 2 = 17`. `a_4 = 3 * a_3 + 2 = 3(17) + 2 = 53`. The sequence is 1, 5, 17, 53, ... D. Translating Between the Two Forms
This is a key skill. It's like translating from Twi to English – expressing the same idea in a different way. From Recursive to Explicit (for AP and GP) Step 1: Look at the recursive formula to identify the pattern and the initial condition. Step 2: If the rule is `a_n = a_(n-1) + d`, you have an AP. Identify `a_1` and `d`. Step 3: Substitute `a_1` and `d` into the explicit AP formula: `a_n = a_1 + (n-1)d`. Step 4: If the rule is `a_n = a_(n-1) * r`, you have a GP. Identify `a_1` and `r`. Step 5: Substitute `a_1` and `r` into the explicit GP formula: `a_n = a_1 * r^(n-1)`.