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

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 2nd Term
Week: 1
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebra
Previous Lesson: .
Topic: SEQUENCE AND SERIES 1 ARITHMETIC PROGRESSION (AP)
Subject Matter: Meaning of sequences indicating first term (a) common difference (d) and the nth term of arithmetic progression and calculating the nth term of an A.P, arithmetic mean and sum of an A.P, practical problem solving involving real life situation on arithmetic mean of an A.P, practical problem solving involving real life situation on sum of A.P

Specific Objectives

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

Cognitive Domain:

• Define a sequence.
• Identify an arithmetic progression (AP).
• State the first term (a) and common difference (d) of an AP.
• Derive and apply the formula for the nth term of an AP.
• Calculate the arithmetic mean between two numbers.
• Derive and apply the formula for the sum of an AP.
• Solve practical problems involving the nth term, arithmetic mean, and sum of an AP.

Affective Domain:

• Participate actively in the derivation of formulas.
• Appreciate the real-life applications of arithmetic progression.

Psychomotor Domain:

• Accurately calculate the nth term and sum of an AP.
• Correctly solve word problems related to AP.

Social Domain:

• Collaborate effectively during problem-solving activities.

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

Instructional Materials

The teacher will teach this lesson with the aid of:

• Ages of students
• Poles and pillars of different height
• Other objects of different sizes
• Numbers written on flashcards

Rationale for the Lesson

This lesson helps pupils understand patterns in numbers and real-life situations. It enables them to predict future values or sums, which is important for planning and problem-solving in various aspects of daily life.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of number patterns, basic algebraic operations, and solving linear equations.

Lesson Content/Board Summary

SEQUENCE AND SERIES: ARITHMETIC PROGRESSION (AP)

1. Meaning of Sequence

A sequence is an ordered list of numbers (or objects) that follows a particular rule or pattern. Each number in the sequence is called a term.

Examples of sequences include:

• 2, 4, 6, 8, … (Even numbers)
• 1, 3, 5, 7, … (Odd numbers)
• 1, 4, 9, 16, … (Square numbers)

2. Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (d).

The general form of an AP is: a, a+d, a+2d, a+3d, …

Where:

• a is the first term.
• d is the common difference (d = T_n – T_n-1).

Example: The sequence 3, 7, 11, 15, … is an AP because the common difference is 4 (7-3=4, 11-7=4, 15-11=4). Here, a=3 and d=4.

3. The nth Term of an A.P. (T_n)

The formula for the nth term of an Arithmetic Progression is given by:

T_n = a + (n-1)d

Where:

• T_n is the nth term.
• a is the first term.
• n is the term number.
• d is the common difference.

Example 1: Finding the nth term

Find the 10th term of the AP: 2, 5, 8, …

Solution:

Given: a = 2, d = 5 – 2 = 3, n = 10

Using the formula T_n = a + (n-1)d

T₁₀ = 2 + (10-1)3

T₁₀ = 2 + (9)3

T₁₀ = 2 + 27

T₁₀ = 29

The 10th term is 29.

Example 2: Finding the first term or common difference

If the 5th term of an AP is 17 and the common difference is 3, find the first term (a).

Solution:

Given: T₅ = 17, d = 3, n = 5

Using the formula T_n = a + (n-1)d

17 = a + (5-1)3

17 = a + (4)3

17 = a + 12

a = 17 – 12

a = 5

The first term is 5.

Example 3: Finding the term number (n)

Which term of the AP: 4, 6, 8, … is 20?

Solution:

Given: a = 4, d = 6 – 4 = 2, T_n = 20

Using the formula T_n = a + (n-1)d

20 = 4 + (n-1)2

20 – 4 = (n-1)2

16 = 2(n-1)

Divide both sides by 2:

8 = n-1

n = 8 + 1

n = 9

The 9th term is 20.

4. Arithmetic Mean (AM)

The arithmetic mean of two numbers is the term that lies exactly between them in an arithmetic progression. For any three terms a, x, b in an AP, x is the arithmetic mean.

The formula for the arithmetic mean (AM) of two numbers ‘a’ and ‘b’ is:

AM = (a + b) / 2

Example 1: Finding the arithmetic mean

Find the arithmetic mean of 10 and 24.

Solution:

Using the formula AM = (a + b) / 2

AM = (10 + 24) / 2

AM = 34 / 2

AM = 17

The arithmetic mean is 17.

Example 2: Inserting arithmetic means

Insert two arithmetic means between 5 and 20.

Solution:

Let the AP be 5, x, y, 20.

Here, a = 5 and T₄ = 20. The number of terms is n = 4.

Using T_n = a + (n-1)d

T₄ = a + (4-1)d

20 = 5 + 3d

20 – 5 = 3d

15 = 3d

d = 15 / 3

d = 5

Now find x and y:

x = a + d = 5 + 5 = 10

y = a + 2d = 5 + 2(5) = 5 + 10 = 15

The two arithmetic means are 10 and 15. The AP is 5, 10, 15, 20.

5. Sum of an Arithmetic Progression (S_n)

The sum of the first ‘n’ terms of an Arithmetic Progression is denoted by S_n.

There are two main formulas for the sum:

Formula 1 (when the last term is unknown):

S_n = n/2 [2a + (n-1)d]

Formula 2 (when the last term ‘l’ is known):

S_n = n/2 [a + l]

Where:

• S_n is the sum of the first n terms.
• a is the first term.
• n is the number of terms.
• d is the common difference.
• l is the last term (T_n).

Example 1: Finding the sum of n terms

Find the sum of the first 12 terms of the AP: 3, 7, 11, …

Solution:

Given: a = 3, d = 7 – 3 = 4, n = 12

Using the formula S_n = n/2 [2a + (n-1)d]

S₁₂ = 12/2 [2(3) + (12-1)4]

S₁₂ = 6 [6 + (11)4]

S₁₂ = 6 [6 + 44]

S₁₂ = 6 [50]

S₁₂ = 300

The sum of the first 12 terms is 300.

Example 2: Finding ‘n’ given the sum

The sum of the first ‘n’ terms of an AP is 120. If the first term (a) is 5 and the common difference (d) is 2, find ‘n’.

Solution:

Given: S_n = 120, a = 5, d = 2

Using the formula S_n = n/2 [2a + (n-1)d]

120 = n/2 [2(5) + (n-1)2]

Multiply both sides by 2:

240 = n [10 + 2n – 2]

240 = n [8 + 2n]

240 = 8n + 2n²

Rearrange into a quadratic equation:

2n² + 8n – 240 = 0

Divide by 2:

n² + 4n – 120 = 0

Factorize the quadratic equation:

(n + 12)(n – 10) = 0

So, n = -12 or n = 10.

Since the number of terms cannot be negative, n = 10.

The number of terms is 10.

6. Practical Problems on Arithmetic Progression

Arithmetic progression is useful in solving real-life problems involving quantities that increase or decrease by a constant amount.

Example 1 (Real-life nth term):

A worker’s starting salary is N50,000 per month. If his salary increases by N5,000 annually, what will be his monthly salary in the 7th year?

Solution:

This is an AP with a = 50,000 (first year’s salary), d = 5,000 (annual increment), n = 7 (7th year).

Using T_n = a + (n-1)d

T₇ = 50,000 + (7-1)5,000

T₇ = 50,000 + (6)5,000

T₇ = 50,000 + 30,000

T₇ = 80,000

His monthly salary in the 7th year will be N80,000.

Example 2 (Real-life Sum):

A cinema has 20 rows of seats. The first row has 15 seats, and each subsequent row has 2 more seats than the row in front of it. How many seats are there in total in the cinema?

Solution:

This is an AP with a = 15 (seats in 1st row), d = 2 (increase per row), n = 20 (number of rows).

We need to find the total sum of seats, S₂₀.

Using the formula S_n = n/2 [2a + (n-1)d]

S₂₀ = 20/2 [2(15) + (20-1)2]

S₂₀ = 10 [30 + (19)2]

S₂₀ = 10 [30 + 38]

S₂₀ = 10 [68]

S₂₀ = 680

There are a total of 680 seats in the cinema.

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 begins by asking pupils to observe patterns in everyday life, such as the arrangement of numbers on a clock face or the growth in height of a plant over days. The teacher then introduces the concept of sequences as ordered lists of numbers.
Pupils’ Activity: Pupils observe and respond to the teacher’s questions about patterns.
Learning Point: Pupils are introduced to the idea of patterns and sequences.

Step 2: Meaning of Sequence

Time: 5 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher formally defines a sequence and provides various examples, asking pupils to identify the rule governing each sequence. Examples include 2, 4, 6, 8… and 1, 4, 9, 16…
Pupils’ Activity: Pupils listen, define sequences, and identify rules for given examples.
Learning Point: Pupils understand what a sequence is and can identify its pattern.

Step 3: Arithmetic Progression (AP)

Time: 7 minutes
Teaching Skill: Concept Development
Teacher’s Activity: The teacher introduces Arithmetic Progression (AP) as a special type of sequence with a constant common difference. The teacher explains and demonstrates how to identify the first term (a) and common difference (d), using examples like 3, 7, 11, 15, … and 20, 18, 16, …
Pupils’ Activity: Pupils define AP, identify ‘a’ and ‘d’ from given sequences, and write out general terms of an AP.
Learning Point: Pupils understand the characteristics of an AP, including ‘a’ and ‘d’.

Step 4: The nth Term of an A.P.

Time: 8 minutes
Teaching Skill: Formula Derivation and Application
Teacher’s Activity: The teacher guides pupils to derive the formula T_n = a + (n-1)d using the general form of an AP. The teacher then works through examples on calculating the nth term, and solving for ‘a’, ‘d’, or ‘n’ when other values are given.
Pupils’ Activity: Pupils participate in deriving the formula and solve example problems on the nth term.
Learning Point: Pupils can use the formula to find any term of an AP or other missing values.

Step 5: Arithmetic Mean (AM)

Time: 6 minutes
Teaching Skill: Definition and Problem Solving
Teacher’s Activity: The teacher defines arithmetic mean and explains its relation to AP. The teacher then presents the formula AM = (a+b)/2 and works through examples, including inserting multiple arithmetic means between two numbers.
Pupils’ Activity: Pupils define AM, apply the formula, and practice inserting means.
Learning Point: Pupils can calculate the arithmetic mean and insert terms into an AP.

Step 6: Sum of an Arithmetic Progression (S_n)

Time: 7 minutes
Teaching Skill: Formula Derivation and Application
Teacher’s Activity: The teacher guides pupils to derive the formulas for the sum of an AP: S_n = n/2 [2a + (n-1)d] and S_n = n/2 [a + l]. The teacher then provides examples on how to calculate the sum and how to find ‘n’ or other values when the sum is given.
Pupils’ Activity: Pupils participate in deriving the formulas and solve problems involving the sum of an AP.
Learning Point: Pupils can calculate the sum of an AP and solve for related missing values.

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 sequence and an Arithmetic Progression (AP).
2. State the formula for the nth term of an AP.
3. Find the 7th term of the AP: 5, 9, 13, …
4. Calculate the arithmetic mean of 15 and 35.
5. State the formula for the sum of the first ‘n’ terms of an AP.
6. Find the sum of the first 10 terms of the AP: 2, 6, 10, …

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 2 minutes
Teaching Skill: Summarizing
Teacher’s Activity: The teacher summarizes the key concepts of sequences, arithmetic progression, nth term, arithmetic mean, and sum of an AP, emphasizing their practical applications. The teacher assigns homework.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Reinforcement of the day’s lesson.

Lesson Keywords

• Sequence – An ordered list of numbers following a rule.
• Arithmetic Progression (AP) – A sequence where the difference between consecutive terms is constant.
• First Term (a) – The initial term of an AP.
• Common Difference (d) – The constant difference between consecutive terms in an AP.
• nth Term (T_n) – The value of the term at a specific position ‘n’ in an AP.
• Arithmetic Mean (AM) – The term exactly between two numbers in an AP.
• Sum of an AP (S_n) – The total sum of the first ‘n’ terms in an AP.

Differentiation

For pupils who grasp concepts quickly, the teacher will provide more complex problems involving simultaneous equations or real-life scenarios requiring multiple steps. For pupils needing more support, the teacher will offer simplified examples, step-by-step guidance, and direct practice on identifying ‘a’ and ‘d’ before moving to formulas.

Note for teachers using this lesson plan

Ensure that pupils understand the conceptual basis of AP before moving to formula application. Encourage active participation in deriving formulas to improve retention. Use visual aids like number lines or physical objects to demonstrate patterns and common differences. Emphasize the real-life relevance of AP to motivate pupils.