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

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: Second Term
Week: 2
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Sequences 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 indicating first term (a) common ratio (r) and nth term of 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 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.
• Calculate the nth term of a given G.P.
• Determine the geometric mean between two numbers.
• State the formulas for the sum of the first n terms of a G.P.
• Calculate the sum of the first n terms of a G.P.
• State the formula for the sum to infinity of a G.P.
• Calculate the sum to infinity of a G.P.

Affective Domain:

• Appreciate the real-life applications of geometrical progression.
• Show interest in solving problems involving sequences and series.

Psychomotor Domain:

• Solve various problems on geometrical progression, including practical problems.
• Accurately apply the derived formulas to solve G.P problems.

Social Domain:

• Collaborate with peers to solve challenging problems related to G.P.

Reference Materials

The following resources were used in planning this lesson:

• 9 Years Basic Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2 by J.B. Channon et al.
• https://www.waecdirect.org/
• https://www.neco.gov.ng/

Instructional Materials

The teacher will teach this lesson with the aid of:

• Charts showing examples of geometrical progressions.
• Poles and pillars of different heights to illustrate growth/decay patterns.
• Objects of different sizes to demonstrate ratios.
• Calculators.

Rationale for the Lesson

This lesson helps pupils understand patterns of growth and decay in numbers and real-life situations. Understanding geometrical progression enables pupils to solve problems related to compound interest, population growth, and depreciation, which are important in everyday life.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of sequences, series, and arithmetic progression.

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).

Examples of G.P. include:

• 2, 4, 8, 16, … (common ratio = 2)
• 81, 27, 9, 3, … (common ratio = 1/3)
• -3, 6, -12, 24, … (common ratio = -2)

First Term (a) and Common Ratio (r)

In a G.P., the first term is denoted by ‘a’. The common ratio ‘r’ is obtained by dividing any term by its preceding term.

If the terms of a G.P. are T₁, T₂, T₃, …, then:

• First term (a) = T₁
• Common ratio (r) = T₂/T₁ = T₃/T₂ = T₄/T₃ = …

Example 1: Find the first term and common ratio of the sequence 3, 6, 12, 24, …

Solution:

Step 1: Identify the first term.

a = 3

Step 2: Calculate the common ratio.

r = 6/3 = 2

r = 12/6 = 2

r = 24/12 = 2

Therefore, the first term (a) is 3 and the common ratio (r) is 2.

Example 2: Find the first term and common ratio of the sequence 100, 50, 25, …

Solution:

Step 1: Identify the first term.

a = 100

Step 2: Calculate the common ratio.

r = 50/100 = 1/2

r = 25/50 = 1/2

Therefore, the first term (a) is 100 and the common ratio (r) is 1/2.

The nth Term of a G.P. (T_n)

The general formula for the nth term of a G.P. is given by:

T_n = ar^n-1

where:

• T_n is the nth term
• a is the first term
• r is the common ratio
• n is the number of terms

Example 1: Find the 5th term of the G.P. 2, 6, 18, …

Solution:

Step 1: Identify the first term (a) and common ratio (r).

a = 2

r = 6/2 = 3

Step 2: Identify the term number (n).

n = 5

Step 3: Apply the formula T_n = ar^n-1.

T₅ = 2 * (3)^5-1

T₅ = 2 * (3)⁴

T₅ = 2 * 81

T₅ = 162

The 5th term is 162.

Example 2: The 3rd term of a G.P. is 48 and the 6th term is 384. Find the first term and the common ratio.

Solution:

Step 1: Write down the given information using the nth term formula.

T₃ = ar³⁻¹ = ar² = 48 (Equation 1)

T₆ = ar⁶⁻¹ = ar⁵ = 384 (Equation 2)

Step 2: Divide Equation 2 by Equation 1 to find ‘r’.

(ar⁵) / (ar²) = 384 / 48

r³ = 8

Step 3: Solve for ‘r’.

r = ³√8

r = 2

Step 4: Substitute ‘r’ into Equation 1 to find ‘a’.

a(2)² = 48

4a = 48

a = 48 / 4

a = 12

The first term (a) is 12 and the common ratio (r) is 2.

Geometric Mean (G.M.)

If a, G, b are three consecutive terms in a G.P., then G is the geometric mean between a and b.

The formula for the geometric mean is:

G = √(ab)

Example 1: Find the geometric mean between 4 and 9.

Solution:

Step 1: Identify a and b.

a = 4, b = 9

Step 2: Apply the formula G = √(ab).

G = √(4 * 9)

G = √36

G = 6

The geometric mean is 6.

Example 2: Insert two geometric means between 3 and 81.

Solution:

Step 1: Let the G.P. be 3, x, y, 81. Here, a = 3 and T₄ = 81.

Step 2: Use the nth term formula to find ‘r’.

T₄ = ar³

81 = 3 * r³

Step 3: Solve for ‘r’.

r³ = 81 / 3

r³ = 27

r = ³√27

r = 3

Step 4: Find the geometric means (x and y).

x = a * r = 3 * 3 = 9

y = x * r = 9 * 3 = 27

The two geometric means are 9 and 27. The sequence is 3, 9, 27, 81.

Sum of Terms of a G.P. (S_n)

The sum of the first ‘n’ terms of a G.P. is given by two formulas, depending on the value of the common ratio ‘r’.

If r > 1:

S_n = a(rⁿ – 1) / (r – 1)

If r < 1 (or -1 < r < 1):

S_n = a(1 – rⁿ) / (1 – r)

where:

• S_n is the sum of the first n terms
• a is the first term
• r is the common ratio
• n is the number of terms

Example 1: Find the sum of the first 6 terms of the G.P. 2, 6, 18, …

Solution:

Step 1: Identify a, r, and n.

a = 2

r = 6/2 = 3

n = 6

Step 2: Since r = 3 (which is > 1), use the formula S_n = a(rⁿ – 1) / (r – 1).

S₆ = 2 * (3⁶ – 1) / (3 – 1)

S₆ = 2 * (729 – 1) / 2

S₆ = 728

The sum of the first 6 terms is 728.

Example 2: Find the sum of the first 5 terms of the G.P. 64, 32, 16, …

Solution:

Step 1: Identify a, r, and n.

a = 64

r = 32/64 = 1/2

n = 5

Step 2: Since r = 1/2 (which is < 1), use the formula S_n = a(1 – rⁿ) / (1 – r).

S₅ = 64 * (1 – (1/2)⁵) / (1 – 1/2)

S₅ = 64 * (1 – 1/32) / (1/2)

S₅ = 64 * (31/32) / (1/2)

S₅ = (64 * 31 / 32) * 2

S₅ = (2 * 31) * 2

S₅ = 62 * 2

S₅ = 124

The sum of the first 5 terms is 124.

Sum to Infinity of a G.P. (S_∞)

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

The formula for the sum to infinity is:

S_∞ = a / (1 – r)

where:

• S_∞ is the sum to infinity
• a is the first term
• r is the common ratio (|r| < 1)

Example 1: Find the sum to infinity of the G.P. 10, 5, 2.5, …

Solution:

Step 1: Identify a and r.

a = 10

r = 5/10 = 1/2

Step 2: Check if |r| < 1. Since |1/2| = 1/2 < 1, the sum to infinity exists.

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

S_∞ = 10 / (1 – 1/2)

S_∞ = 10 / (1/2)

S_∞ = 10 * 2

S_∞ = 20

The sum to infinity is 20.

Example 2: Express the recurring decimal 0.333… as a fraction.

Solution:

Step 1: Write the recurring decimal as a sum of terms in a G.P.

0.333… = 0.3 + 0.03 + 0.003 + …

0.333… = 3/10 + 3/100 + 3/1000 + …

Step 2: Identify a and r for this G.P.

a = 3/10

r = (3/100) / (3/10) = 1/10

Step 3: Since |r| = 1/10 < 1, use the sum to infinity formula.

S_∞ = a / (1 – r)

S_∞ = (3/10) / (1 – 1/10)

S_∞ = (3/10) / (9/10)

S_∞ = (3/10) * (10/9)

S_∞ = 3/9

S_∞ = 1/3

Therefore, 0.333… as a fraction is 1/3.

Practical Problems Involving G.P.

Geometrical progression can be applied to solve real-life problems such as:

• Population growth or decay.
• Compound interest.
• Depreciation of assets.
• Radioactive decay.
• Bouncing ball problems.

Example: A man buys a car for N2,000,000. If the value of the car depreciates by 10% each year, what will be its value after 3 years?

Solution:

Step 1: Identify the initial value (a) and the depreciation rate.

Initial value (a) = N2,000,000

Depreciation rate = 10% = 0.1

Step 2: Calculate the common ratio (r).

If it depreciates by 10%, it retains 100% – 10% = 90% of its value each year.

r = 90% = 0.9

Step 3: Determine the term needed. After 1 year, it’s the 2nd term (ar). After 3 years, it’s the 4th term (ar³).

n = 3 + 1 = 4 (because the initial value is the first term, and after 3 years means 3 depreciation cycles)

Step 4: Apply the nth term formula T_n = ar^n-1.

T₄ = 2,000,000 * (0.9)⁴⁻¹

T₄ = 2,000,000 * (0.9)³

T₄ = 2,000,000 * 0.729

T₄ = 1,458,000

The value of the car after 3 years will be N1,458,000.

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 reminds pupils about arithmetic progression (A.P.) from the previous lesson and asks them to state how terms are generated in an A.P. The teacher then introduces G.P. by presenting a sequence like 2, 4, 8, 16, … and asks pupils to identify the relationship between consecutive terms.
Pupils’ Activity: Pupils recall A.P. and observe the pattern in the new sequence, noting that each term is multiplied by a constant factor.
Learning Point: Pupils are prepared for the new topic by connecting it to prior knowledge and identifying a new type of sequence.

Step 2: Meaning of G.P., First Term, and Common Ratio

Time: 10 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher formally defines geometrical progression and explains how to identify the first term (a) and calculate the common ratio (r) by dividing any term by its preceding term. The teacher provides several examples and asks pupils to find ‘a’ and ‘r’.
Pupils’ Activity: Pupils listen attentively, define G.P., and practice identifying ‘a’ and ‘r’ from given sequences.
Learning Point: Pupils understand the fundamental concepts of G.P., including its definition, first term, and common ratio.

Step 3: The nth Term of a G.P.

Time: 10 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 listing the first few terms (a, ar, ar², ar³, …). The teacher then works through examples on calculating the nth term and finding ‘a’ or ‘r’ when given specific terms.
Pupils’ Activity: Pupils participate in the derivation, copy the formula, and solve example problems on calculating the nth term.
Learning Point: Pupils can state and apply the formula for the nth term of a G.P. to solve problems.

Step 4: Geometric Mean (G.M.)

Time: 5 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the concept of geometric mean between two numbers and presents the formula G = √(ab). The teacher demonstrates how to calculate the geometric mean and how to insert geometric means between two given numbers.
Pupils’ Activity: Pupils listen, copy the formula, and practice calculating geometric means.
Learning Point: Pupils can find the geometric mean between two numbers and insert terms into a G.P.

Step 5: Sum of Terms of a G.P. (S_n)

Time: 10 minutes
Teaching Skill: Formula Application/Problem Solving
Teacher’s Activity: The teacher introduces the two formulas for the sum of the first n terms of a G.P., explaining when to use each (r > 1 or r < 1). The teacher works through examples, guiding pupils on how to apply the correct formula based on the common ratio. Pupils’ Activity: Pupils copy the formulas and work through examples, practicing the calculation of the sum of the first n terms.
Learning Point: Pupils can correctly apply the appropriate formula to calculate the sum of the first n terms of a G.P.

Step 6: Sum to Infinity of a G.P. (S_∞) and Practical Problems

Time: 10 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher explains the condition for the existence of the sum to infinity (|r| < 1) and introduces the formula S_∞ = a / (1 – r). The teacher demonstrates its application, including converting recurring decimals to fractions, and then introduces practical problems involving G.P.
Pupils’ Activity: Pupils understand the concept of sum to infinity, copy the formula, solve related problems, and attempt practical problems.
Learning Point: Pupils can calculate the sum to infinity of a G.P. and apply G.P. concepts to real-life situations.

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. State the formula for the nth term of a G.P.
3. Given the G.P. 5, 10, 20, …, find its common ratio and the 4th term.
4. Calculate the geometric mean between 5 and 20.
5. Find the sum of the first 4 terms of the G.P. 5, 10, 20, …
6. What is the condition for the sum to infinity of a G.P. to exist?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teacher’s Activity: The teacher summarizes the key points of the lesson, emphasizing the formulas for nth term, sum of n terms, geometric mean, and sum to infinity. The teacher then assigns homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils consolidate their understanding of G.P. concepts and are given practice tasks.

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.
• First Term (a) – The initial term of a sequence.
• Common Ratio (r) – The fixed non-zero number by which each term is multiplied to get the next term in a G.P.
• nth Term (T_n) – The value of the term at a specific position ‘n’ in a sequence.
• Geometric Mean (G.M.) – The middle term of three consecutive terms in a G.P., found by √(ab).
• 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 G.P., which exists only when the common ratio |r| < 1.

Differentiation

For pupils who grasp concepts quickly, additional challenging problems involving word problems or deriving formulas can be provided. For pupils who need more support, the teacher will provide simplified examples, step-by-step guidance, and peer tutoring opportunities.

Note for teachers using this lesson plan

Ensure that pupils clearly understand the difference between Arithmetic Progression and Geometrical Progression. Emphasize the importance of identifying ‘a’ and ‘r’ correctly as the first step in solving any G.P. problem. Encourage the use of calculators for complex calculations but stress understanding the formulas and steps. Real-life examples help in making the topic relatable.