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

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: First Term
Week: 6
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebra
Previous Lesson: Approximation: Rounding and Percentage Error.
Topic: SEQUENCE AND SERIES 1 ARITHMETIC PROGRESSION (AP)
Subject Matter: Meaning of sequences, first term (a), common difference (d), the nth term of an A.P, arithmetic mean, sum of an A.P, and practical problem solving involving A.P.

Specific Objectives

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

Cognitive Domain:

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

Affective Domain:

• Participate actively in the derivation of formulas for A.P.
• Appreciate the relevance of arithmetic progression in real-life situations.

Psychomotor Domain:

• Accurately apply the formulas for the nth term, arithmetic mean, and sum of an A.P to solve problems.
• Construct simple arithmetic sequences given specific conditions.

Social Domain:

• Collaborate with peers to discuss and solve problems related to A.P.

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 School 2
• https://www.waeconline.org.ng/curriculum
• https://www.education.gov.ng

Instructional Materials

The teacher will teach this lesson with the aid of:

• Whiteboard/blackboard and markers/chalk
• Textbooks (e.g., New General Mathematics for SS2)
• Chart showing examples of sequences and arithmetic progressions
• Objects of different sizes (e.g., ages of students, poles, pillars) to illustrate sequences

Rationale for the Lesson

This lesson helps pupils understand patterns in numbers, which is a fundamental concept in mathematics. It enables them to solve real-world problems involving quantities that increase or decrease by a constant amount, such as calculating savings over time or determining the height of stacked objects.

Prerequisite/Previous Knowledge

Pupils should have basic knowledge of number patterns, algebraic expressions, and solving simple equations.

Lesson Content/Board Summary

Arithmetic Progression (A.P)

Meaning of Sequence

A sequence is a set of numbers arranged in a definite order according to a specific rule. Each number in the sequence is called a term.

Examples:

• 2, 4, 6, 8, … (rule: add 2 to the previous term)
• 1, 4, 9, 16, … (rule: square the term number)
• 10, 7, 4, 1, … (rule: subtract 3 from the previous term)

Arithmetic Progression (A.P)

An Arithmetic Progression (A.P.) is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is called the common difference (d).

The first term of an A.P. is denoted by ‘a’.

Examples of A.P.:

• 3, 6, 9, 12, … (a = 3, d = 3)
• 20, 18, 16, 14, … (a = 20, d = -2)
• 5, 10, 15, 20, … (a = 5, d = 5)

To find the common difference (d), subtract any term from its succeeding term: d = T₂ – T₁ = T₃ – T₂, and so on.

The nth Term of an A.P.

The formula for the nth term (T_n) 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 number of terms
• d is the common difference

Worked Example 1: Find the 10th term of the A.P. 2, 5, 8, 11, …

Step 1: Identify ‘a’ and ‘d’.
a = 2
d = 5 – 2 = 3

Step 2: State ‘n’.
n = 10

Step 3: Apply the formula T_n = a + (n – 1)d.
T₁₀ = 2 + (10 – 1)3
T₁₀ = 2 + (9)3
T₁₀ = 2 + 27
T₁₀ = 29

Worked Example 2: The 5th term of an A.P. is 16 and the 12th term is 37. Find the first term and the common difference.

Step 1: Formulate equations from the given information.
T₅ = a + (5 – 1)d = a + 4d = 16 (Equation 1)
T₁₂ = a + (12 – 1)d = a + 11d = 37 (Equation 2)

Step 2: Solve the simultaneous equations.
Subtract Equation 1 from Equation 2:
(a + 11d) – (a + 4d) = 37 – 16
7d = 21
d = 21 / 7
d = 3

Step 3: Substitute ‘d’ into Equation 1 to find ‘a’.
a + 4(3) = 16
a + 12 = 16
a = 16 – 12
a = 4

Therefore, the first term (a) is 4 and the common difference (d) is 3.

Arithmetic Mean

The arithmetic mean (A.M.) of two numbers ‘x’ and ‘y’ is the number ‘A’ such that x, A, y form an A.P. It is calculated as:

A = (x + y) / 2

If there are several arithmetic means to be inserted between two numbers, they all form an A.P.

Worked Example 1: Find the arithmetic mean of 10 and 24.

Step 1: Identify ‘x’ and ‘y’.
x = 10, y = 24

Step 2: Apply the formula A = (x + y) / 2.
A = (10 + 24) / 2
A = 34 / 2
A = 17

Worked Example 2: Insert two arithmetic means between 7 and 22.

Step 1: Let the means be A₁ and A₂. The sequence is 7, A₁, A₂, 22.
Here, a = 7 and T₄ = 22.

Step 2: Use T_n = a + (n – 1)d to find ‘d’.
T₄ = a + (4 – 1)d
22 = 7 + 3d
15 = 3d
d = 5

Step 3: Calculate the means.
A₁ = a + d = 7 + 5 = 12
A₂ = A₁ + d = 12 + 5 = 17

The two arithmetic means are 12 and 17.

Sum of an A.P (S_n)

The sum of the first ‘n’ terms of an arithmetic progression (S_n) can be calculated using one of the following formulas:

S_n = n/2 [2a + (n – 1)d]

OR

S_n = n/2 [a + L] (where L is the last term, L = T_n)

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)

Worked Example 1: Find the sum of the first 15 terms of the A.P. 3, 7, 11, 15, …

Step 1: Identify ‘a’, ‘d’, and ‘n’.
a = 3
d = 7 – 3 = 4
n = 15

Step 2: Apply the formula S_n = n/2 [2a + (n – 1)d].
S₁₅ = 15/2 [2(3) + (15 – 1)4]
S₁₅ = 15/2 [6 + (14)4]
S₁₅ = 15/2 [6 + 56]
S₁₅ = 15/2 [62]
S₁₅ = 15 × 31
S₁₅ = 465

Worked Example 2: The first term of an A.P. is 5, and the last term is 45. If there are 11 terms in the A.P., find the sum of the terms.

Step 1: Identify ‘a’, ‘L’, and ‘n’.
a = 5
L = 45
n = 11

Step 2: Apply the formula S_n = n/2 [a + L].
S₁₁ = 11/2 [5 + 45]
S₁₁ = 11/2 [50]
S₁₁ = 11 × 25
S₁₁ = 275

Practical Problem Solving

Arithmetic progressions are used to model situations where quantities change by a constant amount. For example, calculating simple interest, finding the total number of seats in an auditorium with rows of increasing size, or determining the total distance covered in a specific pattern of movement.

Example: A man saves N500 in the first month, N600 in the second month, N700 in the third month, and so on. How much will he save in the 12th month?

This is an A.P. with a = N500, d = N100, and n = 12. We need to find T₁₂.

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 identify patterns in a given sequence of numbers (e.g., 1, 3, 5, 7, … or 20, 15, 10, 5, …).
Pupils’ Activity: Pupils observe the sequences and state the rule governing each pattern.
Learning Point: Pupils recall prior knowledge of number patterns and are introduced to the concept of sequences.

Step 2: Meaning of Sequence and Arithmetic Progression (A.P)

Time: 7 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines a sequence and an arithmetic progression, explaining the terms ‘first term (a)’ and ‘common difference (d)’. The teacher provides various examples of A.P.
Pupils’ Activity: Pupils listen, define sequences and A.P., and identify ‘a’ and ‘d’ from the examples given by the teacher.
Learning Point: Pupils understand the definitions of sequence, A.P., first term, and common difference.

Step 3: Derivation of the nth Term of an A.P.

Time: 7 minutes
Teaching Skill: Guided Discovery/Demonstration
Teacher’s Activity: The teacher guides pupils to observe the relationship between the term number and the number of common differences added to the first term (T₁ = a, T₂ = a+d, T₃ = a+2d, …). This leads to the derivation of T_n = a + (n-1)d.
Pupils’ Activity: Pupils participate in the derivation, observing the pattern and contributing to forming the general formula.
Learning Point: Pupils understand how the formula for the nth term is derived.

Step 4: Calculation of the nth Term of an A.P.

Time: 7 minutes
Teaching Skill: Explanation/Problem Solving
Teacher’s Activity: The teacher explains how to use the nth term formula and works through examples on the board, showing step-by-step calculations.
Pupils’ Activity: Pupils copy the examples and solve similar problems presented by the teacher.
Learning Point: Pupils can apply the formula T_n = a + (n-1)d to calculate specific terms of an A.P.

Step 5: Arithmetic Mean

Time: 4 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher defines arithmetic mean and demonstrates how to calculate it for two numbers and how to insert means between numbers.
Pupils’ Activity: Pupils listen, take notes, and solve simple problems on finding arithmetic means.
Learning Point: Pupils learn to calculate the arithmetic mean.

Step 6: Sum of an A.P (S_n)

Time: 7 minutes
Teaching Skill: Explanation/Formula Application
Teacher’s Activity: The teacher introduces the formulas for the sum of an A.P (S_n = n/2 [2a + (n-1)d] or S_n = n/2 [a + L]) and explains each variable. The teacher solves examples involving the sum of an A.P.
Pupils’ Activity: Pupils write down the formulas, follow the worked examples, and attempt practice questions.
Learning Point: Pupils can apply the formulas for the sum of an A.P. to solve problems.

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. What is an arithmetic progression (A.P.)?
2. Identify the first term (a) and common difference (d) for the A.P.: 5, 8, 11, 14, …
3. State the formula for the nth term of an A.P.
4. Find the 8th term of the A.P.: 3, 7, 11, …
5. Calculate the arithmetic mean of 15 and 35.
6. State the formula for the sum of the first ‘n’ terms of an A.P.
7. Find the sum of the first 10 terms of an A.P. whose first term is 2 and common difference is 4.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 3 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key concepts covered: meaning of sequence and A.P., formulas for nth term, arithmetic mean, and sum of A.P. The teacher assigns relevant homework.
Pupils’ Activity: Pupils listen to the summary and copy the assigned homework.
Learning Point: Pupils consolidate their understanding of the topic and prepare for further practice.

Lesson Keywords

• Sequence – A set of numbers arranged in a definite order.
• Arithmetic Progression (A.P.) – A sequence where the difference between consecutive terms is constant.
• First Term (a) – The initial term of an A.P.
• Common Difference (d) – The constant difference between consecutive terms in an A.P.
• nth Term (T_n) – The term at any given position ‘n’ in an A.P.
• Arithmetic Mean – The middle term of an A.P. of three terms, or the average of a set of numbers.
• Sum of an A.P. (S_n) – The total value obtained by adding the terms of an A.P.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex word problems involving A.P. or introduce challenges like finding the number of terms or the common difference when the sum and other parameters are given. For pupils who need more support, the teacher will provide additional guided practice with simpler examples, use visual aids more extensively, and pair them with stronger pupils for peer learning.

Note for teachers using this lesson plan

Ensure that pupils understand the concept of a common difference before moving to formulas. Emphasize the derivation of the nth term formula to help them understand its logic rather than just memorizing it. Encourage pupils to relate the concepts of A.P. to everyday situations to enhance their understanding and appreciation of mathematics.