Lesson Note on Geometrical Progression: Nth Term, Mean and Sum for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 7
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Sequence and Series
Previous Lesson: Arithmetic Progression: Nth Term, Mean and Sum.
Topic: SEQUENCE AND SERIES (II) GEOMETRICAL PROGRESSION (G.P)
Subject Matter: Meaning of geometrical progression (G.P) indicating first term (a) common ratio (r) and nth term of a G.P and calculation of nth term of G.P, geometric mean and sum of terms of G.P, sum of infinity of G.P, practical problem involving real life situation on G.P

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Define a geometrical progression (G.P).
• Identify the first term (a) and common ratio (r) of a G.P.
• State the formula for the nth term of a G.P.
• State the formulas for the sum of n terms of a G.P.
• State the formula for the sum to infinity of a G.P.

Affective Domain:

• Appreciate the application of G.P in real-life situations.
• Show interest in solving problems involving G.P.

Psychomotor Domain:

• Calculate the nth term of a given G.P.
• Determine the geometric mean between two numbers.
• Calculate the sum of a given number of terms of a G.P.
• Calculate the sum to infinity of a G.P with a common ratio between -1 and 1.
• Solve practical problems involving G.P.

Social Domain:

• Collaborate with peers to solve problems on geometrical progression.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2

Instructional Materials

The teacher will teach this lesson with the aid of:

• Financial reports and budget
• Population figures
• Logarithm table
• Calculator
• Data from school records, health sector, economy etc.

Rationale for the Lesson

Understanding geometrical progression helps pupils to solve problems involving exponential growth or decay, such as population growth, compound interest, and depreciation. It provides a foundation for more advanced mathematical concepts and their applications in various fields.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of arithmetic progression (A.P), basic algebra, and sequences.

Lesson Content/Board Summary

Geometrical Progression (G.P)

Meaning of Geometrical Progression (G.P)

A geometrical progression (G.P) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

Example: 2, 4, 8, 16, … (common ratio r = 2)

The first term is denoted by ‘a’.

The nth Term of a G.P

The formula for the nth term (T_n) of a G.P is given by:

T_n = ar^(n-1)

Where:

• a = the first term
• r = the common ratio
• n = the term number

Example 1: Find the 5th term of the G.P: 3, 6, 12, …

Step 1: Identify ‘a’ and ‘r’.
a = 3
r = 6/3 = 2

Step 2: State the formula for the nth term.
T_n = ar^(n-1)

Step 3: Substitute the values for n=5.
T₅ = 3 * 2^(5-1)
T₅ = 3 * 2⁴
T₅ = 3 * 16
T₅ = 48

Geometric Mean

The geometric mean (GM) of two numbers ‘x’ and ‘y’ is given by:

GM = √(xy)

If a, GM, b are in G.P, then GM/a = b/GM, which implies GM² = ab, so GM = √(ab).

Example 2: Find the geometric mean between 4 and 16.

Step 1: Identify ‘x’ and ‘y’.
x = 4
y = 16

Step 2: State the formula for geometric mean.
GM = √(xy)

Step 3: Substitute the values.
GM = √(4 * 16)
GM = √64
GM = 8

Sum of n Terms of a G.P

The sum of the first n terms (S_n) of a G.P is given by two formulas:

1. If r > 1:

S_n = a(rⁿ – 1) / (r – 1)

2. If r < 1:

S_n = a(1 – rⁿ) / (1 – r)

Example 3: Find the sum of the first 4 terms of the G.P: 2, 6, 18, …

Step 1: Identify ‘a’, ‘r’, and ‘n’.
a = 2
r = 6/2 = 3
n = 4

Step 2: Choose the appropriate formula (r = 3 > 1).
S_n = a(rⁿ – 1) / (r – 1)

Step 3: Substitute the values.
S₄ = 2(3⁴ – 1) / (3 – 1)
S₄ = 2(81 – 1) / 2
S₄ = 2(80) / 2
S₄ = 80

Sum to Infinity of a G.P

The sum to infinity (S_∞) of a G.P exists only if the absolute value of the common ratio is less than 1 (i.e., |r| < 1 or -1 < r < 1).

The formula for the sum to infinity is:

S_∞ = a / (1 – r)

Example 4: Find the sum to infinity of the G.P: 8, 4, 2, …

Step 1: Identify ‘a’ and ‘r’.
a = 8
r = 4/8 = 1/2

Step 2: Check the condition for sum to infinity (|r| < 1).
|1/2| < 1, so the sum to infinity exists.

Step 3: State the formula.
S_∞ = a / (1 – r)

Step 4: Substitute the values.
S_∞ = 8 / (1 – 1/2)
S_∞ = 8 / (1/2)
S_∞ = 8 * 2
S_∞ = 16

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher reviews the previous lesson on Arithmetic Progression (A.P) and asks pupils to state the characteristics of an A.P. The teacher then introduces the concept of a geometrical progression as another type of sequence.
Pupils’ Activity: Pupils recall and state the characteristics of an A.P. They listen attentively to the introduction of G.P.
Learning Point: Pupils link prior knowledge to the new topic.

Step 2: Meaning of Geometrical Progression and Common Ratio

Time: 10 minutes
Teaching Skill: Explanation/Illustration
Teacher’s Activity: The teacher defines geometrical progression (G.P) and explains how to find the common ratio (r) by dividing any term by its preceding term. The teacher provides examples like 2, 4, 8, 16… and 27, 9, 3, 1…
Pupils’ Activity: Pupils define G.P and identify the common ratio in given sequences. They also provide their own examples.
Learning Point: Pupils understand what a G.P is and how to identify its common ratio.

Step 3: The nth Term of a G.P

Time: 8 minutes
Teaching Skill: Derivation/Problem Solving
Teacher’s Activity: The teacher guides pupils to derive the formula for the nth term (T_n = ar^(n-1)) by observing the pattern in a G.P (a, ar, ar², ar³,…). The teacher then solves an example using the formula.
Pupils’ Activity: Pupils participate in the derivation and copy the formula. They work through the example provided by the teacher.
Learning Point: Pupils can state and use the formula for the nth term of a G.P.

Step 4: Geometric Mean

Time: 5 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the concept of geometric mean between two numbers and derives the formula GM = √(ab). The teacher then demonstrates how to calculate the geometric mean between two given numbers.
Pupils’ Activity: Pupils understand the concept and formula for geometric mean. They practice calculating geometric means.
Learning Point: Pupils can determine the geometric mean between two numbers.

Step 5: Sum of n Terms of a G.P

Time: 7 minutes
Teaching Skill: Explanation/Problem Solving
Teacher’s Activity: The teacher introduces the formulas for the sum of n terms (S_n) for both r > 1 and r < 1, explaining the conditions for each. The teacher solves an example for each case.
Pupils’ Activity: Pupils copy the formulas and pay attention to the conditions. They follow the worked examples and attempt similar problems.
Learning Point: Pupils can apply the correct formula to find the sum of n terms of a G.P.

Step 6: Sum to Infinity of a G.P

Time: 5 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher explains when a sum to infinity exists (|r| < 1) and presents the formula S_∞ = a / (1 – r). The teacher solves an example demonstrating its application.
Pupils’ Activity: Pupils understand the condition for sum to infinity and the formula. They practice solving problems involving sum to infinity.
Learning Point: Pupils can calculate the sum to infinity for a convergent G.P.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define a Geometrical Progression (G.P).
2. How do you find the common ratio of a G.P?
3. State the formula for the nth term of a G.P.
4. Find the 4th term of the G.P: 5, 10, 20, …
5. Calculate the sum of the first 3 terms of the G.P: 3, 9, 27, …
6. When does the sum to infinity of a G.P exist?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teacher’s Activity: The teacher summarizes the key points of the lesson on G.P, emphasizing the formulas for nth term, sum of n terms, and sum to infinity. The teacher assigns homework involving practical problems related to G.P.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils consolidate their learning and prepare for further practice.

Lesson Keywords

• Geometrical Progression (G.P) – A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number.
• Common Ratio (r) – The fixed number by which each term is multiplied to get the next term in a G.P.
• First Term (a) – The initial term in a sequence.
• nth Term (T_n) – The term at any given position ‘n’ in a sequence.
• Geometric Mean (GM) – The central term in a G.P of three terms, found by taking the square root of the product of the two outer terms.
• Sum of n Terms (S_n) – The total sum of the first ‘n’ terms of a G.P.
• Sum to Infinity (S_∞) – The sum of an infinite number of terms of a G.P, which exists when the common ratio is between -1 and 1.

Differentiation

For pupils who grasp concepts quickly, the teacher will provide more complex problems involving deriving unknown terms or common ratios from given information. For pupils needing more support, the teacher will provide additional guided practice with simpler G.P sequences and focus on step-by-step problem-solving for each formula.

Note for teachers using this lesson plan

Ensure that pupils clearly understand the conditions for using each sum formula (r > 1 vs. r < 1) and the condition for the sum to infinity (|r| < 1). Encourage pupils to show all working steps for calculations to avoid errors and enhance understanding.