Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Sequence
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Sequence
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: Further Mathematics
Class: Senior Secondary 1
Term: 3rd Term
Week: 2
Theme: Pure Mathematics
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 topic introduces students to the fundamental concept of sequences, which are ordered lists of numbers or objects that follow a specific pattern or rule. Understanding sequences is crucial for developing logical reasoning and problem-solving skills, and it forms the foundation for more advanced topics in mathematics such as series, functions, and calculus. In real life, sequences help model and predict various phenomena, from population growth to financial investments and even the structured patterns found in traditional Nigerian art and textiles.
Specific Performance Objectives:
Definition of a Sequence: A sequence is an ordered list of numbers or objects that follows a specific rule or pattern. Each number in the sequence is called a term. The terms are usually separated by commas and enclosed in curly braces, though often just listed.
Example 1: 2, 4, 6, 8, ... (sequence of even numbers)
Example 2: 1, 3, 5, 7, ... (sequence of odd numbers)
Example 3: 1, 4, 9, 16, ... (sequence of perfect squares)
Terms of a Sequence: Each number in a sequence is called a term. These terms are often denoted by $T_n$ or $a_n$, where 'n' represents the position of the term in the sequence. $T_1$ (or $a_1$) is the first term. $T_2$ (or $a_2$) is the second term. $T_n$ (or $a_n$) is the nth term or general term, which is a formula that allows one to find any term in the sequence given its position 'n'.
Finding the nth Term of a Sequence: The core task is to identify the rule that generates the sequence and express it as a formula for $T_n$. This often involves observing the relationship between the term number (n) and the term value ($T_n$).
Common Types of Patterns to Look For:
1. Arithmetic Progression (AP): Each term is obtained by adding a constant value (common difference, d) to the previous term. (Though APs are a separate topic, recognizing a constant difference is the first step).
Example: 3, 7, 11, 15, ... (Here, $d=4$. $T_n = a + (n-1)d$)
2. Geometric Progression (GP): Each term is obtained by multiplying the previous term by a constant value (common ratio, r). (Similarly, GPs are a separate topic, but recognizing a constant ratio is useful).
Example: 2, 6, 18, 54, ... (Here, $r=3$. $T_n = ar^{n-1}$)
3. Quadratic Sequences: The second differences between consecutive terms are constant. (The formula for $T_n$ will involve $n^2$).
4. Cubic Sequences: The third differences between consecutive terms are constant. (The formula for $T_n$ will involve $n^3$).
5. Exponential Sequences: The terms involve powers of a constant base ($a^n$ or $a^{n-1}$).
6. Sequences involving fractions, alternating signs, or combinations of the above. Step-by-step Reasoning for Finding the nth Term:
1. List the terms and their positions: Create a table or simply list the terms ($T_n$) alongside their corresponding position (n). n: 1, 2, 3, 4, ... $T_n$: $T_1$, $T_2$, $T_3$, $T_4$, ...
2. Calculate the differences between consecutive terms (first differences): $T_2 - T_1$, $T_3 - T_2$, $T_4 - T_3$, etc.
3. Observe the pattern in the first differences: If the first differences are constant, the sequence is arithmetic, and the general term is linear ($T_n = an + b$). The constant difference is 'a'. If the first differences are not constant, calculate the second differences ($D_2 - D_1$, $D_3 - D_2$, etc.).
4. Observe the pattern in the second differences: If the second differences are constant, the sequence is quadratic ($T_n = an^2 + bn + c$). The constant second difference is equal to $2a$. So, $a = (\text{constant second difference}) / 2$. If not constant, calculate third differences, and so on.
5. Alternatively, look for other relationships: Is each term a multiple of 'n'? (e.g., $T_n = 3n$) Is each term a square, cube, or power of 'n' or ($n \pm k$)? (e.g., $T_n = n^2$, $T_n = (n+1)^2$) Is there a constant ratio between terms? (e.g., $T_n = 2^n$) Does the pattern alternate (e.g., positive/negative terms)? Worked
Examples: Example 1: Finding the nth term of a linear sequence. A farmer in Kano observes that the number of yam seedlings planted in successive rows follows a pattern: 5, 8, 11, 14, ... Find the formula for the nth term, $T_n$.
Solution:*
1. List terms and positions: n: 1, 2, 3, 4 $T_n$: 5, 8, 11, 14
2. Calculate first differences: $8 - 5 = 3$ $11 - 8 = 3$ $14 - 11 = 3$
3. The first differences are constant (3). This indicates a linear sequence of the form $T_n = an + b$. Here, $a of a linear sequence. A farmer in Kano observes that the number of yam seedlings planted in successive rows follows a pattern: 5, 8, 11, 14, ... Find the formula for the nth term, $T_n$.
Solution:
1. List terms and positions: n: 1, 2, 3, 4 $T_n$: 5, 8, 11, 14
2. Calculate first differences: $8 - 5 = 3$ $11 - 8 = 3$ $14 - 11 = 3$
3. The first differences are constant (3). This indicates a linear sequence of the form $T_n = an + b$. Here, $a = 3$. So, $T_n = 3n + b$.
4. Substitute the value of $T_1$ (n=1, $T_n$=5) into the formula to find 'b': $5 = 3(1) + b$ $5 = 3 + b$ $b = 5 - 3 = 2$
5. Therefore, the formula for the nth term is $T_n = 3n + 2$.
Verification: $T_1 = 3(1) + 2 = 5$ $T_2 = 3(2) + 2 = 8$ $T_3 = 3(3) + 2 = 11$ The formula is correct.
Example 2: Finding the nth term of a quadratic sequence.
A sequence is given as: 2, 6, 12, 20, 30, ... Find the formula for the nth term, $T_n$.
Solution:
1. List terms and positions: n: 1, 2, 3, 4, 5 $T_n$: 2, 6, 12, 20, 30
2. Calculate first differences: $6 - 2 = 4$ $12 - 6 = 6$ $20 - 12 = 8$ $30 - 20 = 10$ (First differences: 4, 6, 8, 10 - not constant)
3. Calculate second differences: $6 - 4 = 2$ $8 - 6 = 2$ $10 - 8 = 2$ (Second differences: 2, 2, 2 - constant)
4. Since the second differences are constant, the sequence is quadratic, with the form $T_n = an^2 + bn + c$. The constant second difference is $2a$. So, $2a = 2 \Rightarrow a = 1$. Substitute $a=1$ into the general form. Now $T_n = n^2 + bn + c$. Use $T_1$ (n=1, $T_n$=2): $2 = (1)^2 + b(1) + c$ $2 = 1 + b + c \Rightarrow b + c = 1$ (Equation 1) Use $T_2$ (n=2, $T_n$=6): $6 = (2)^2 + b(2) + c$ $6 = 4 + 2b + c \Rightarrow 2b + c = 2$ (Equation 2)
5. Solve the system of equations for b and c: (Equation 2) - (Equation 1): $(2b + c) - (b + c) = 2 - 1$ $b = 1$
6. Substitute $b=1$ into Equation 1: $1 + c = 1 \Rightarrow c = 0$
7. Therefore, the formula for the nth term is $T_n = 1n^2 + 1n + 0$, which simplifies to $T_n = n^2 + n$.
Verification: $T_1 = 1^2 + 1 = 2$ $T_2 = 2^2 + 2 = 6$ $T_3 = 3^2 + 3 = 12$ The formula is correct.
Example 3: Finding the nth term involving multiplication or powers.** Consider the sequence of terms representing the number of branches on a specific plant type after n weeks, starting from week 1: 3, 9, 27, 81, ... Find the formula for the nth term, $T_n$.
Solution:
1. List terms and positions: n: 1, 2, 3, 4 $T_n$: 3, 9, 27, 81
2. Observe the relationship: $T_1 = 3 = 3^1$ $T_2 = 9 = 3^2$ $T_3 = 27 = 3^3$ $T_4 = 81 = 3^4$
3. It is clear that each term is 3 raised to the power of its position 'n'.
4. Therefore, the formula for the nth term is $T_n = 3^n$.
Materials: Whiteboard, markers, chalk, rulers, grid paper (optional), worksheets.
Introduction (10 minutes): Teacher Activity: Begin by writing simple number patterns on the board and asking students to identify the next two or three numbers.
Example 1: 1, 2, 3, 4, __, __ Example 2: 5, 10, 15, 20, __, __ Example 3: 1, 4, 9, 16, __, __ Student Activity: Students actively participate by shouting out the next numbers and briefly explaining the pattern they observed. Concept Definition and Exploration (15 minutes): Teacher Activity: Explain that these ordered lists of numbers are called "sequences." Formally define a sequence and explain what a "term" is, using $T_n$ or $a_n$ notation. Provide diverse examples of sequences, including some from Nigerian contexts (e.g., weekly savings increasing by a fixed amount, number of people in a family tree assuming a certain branching pattern). Emphasize that a key aspect of sequences is having a "rule" or "pattern." Student Activity: Students write down the definition of a sequence and term in their notebooks. Students identify the first, second, and third terms of given sequences. Students volunteer to create their own simple sequences following a clear rule.
Developing the nth Term (40 minutes): Teacher Activity: Introduce the concept of finding the nth term as a way to describe the rule of the sequence using algebra. Guide students through the process of finding the nth term for linear sequences first (constant first difference). Use Example 1 from Key Concepts. Demonstrate how to calculate first and second differences. Introduce quadratic sequences (constant second difference) using Example
2. Clearly show the derivation of $a$, $b$, and $c$. Present examples of sequences with multiplicative or exponential patterns (like Example 3), encouraging students to look for ratios or powers. Facilitate a short group discussion on various strategies for identifying patterns (e.g., addition/subtraction, multiplication/division, squaring/cubing).
Student Activity: Students work in small groups (2-3 students) to identify patterns and find the nth term for several practice sequences provided by the teacher on worksheets or the board. Groups present their findings and methods to the class. Students practice substituting values of 'n' into derived formulas to verify their accuracy.
Consolidation and Wrap-up (5 minutes): Teacher Activity: Recap the definition of a sequence and the importance of finding the nth term. Address any remaining questions.
Student Activity: Students ask clarifying questions and summarize the main learning points in their own words. The teacher should present these questions on the board or as a handout and guide students through solving them step-by-step, encouraging discussion and different approaches.
Question 1: Define a sequence. Give two examples of sequences found in your daily life or observed in your community.
Solution: A sequence is an ordered list of numbers, objects, or events that follow a specific pattern or rule. Each item in the list is called a term.
Example 1 (Daily Life): The number of bottles of soft drink a hawker sells daily for a week, assuming a pattern (e.g., 20, 25, 30, 35, 40, 45, 50).
Example 2 (Community): The layers of bricks laid on a building project each day, assuming progress is sequential (e.g., 100, 200, 300, 400, ...).
Question 2: Consider the sequence: 7, 12, 17, 22, ... a) Identify the first three terms of this sequence. b) What is the common difference? c) Find the 10th term of the sequence.
Solution: a) The first three terms are 7, 12, and 17. b) Calculate the differences between consecutive terms: $12 - 7 = 5$ $17 - 12 = 5$ $22 - 17 = 5$ The common difference is 5. c) To find the 10th term, first find the general formula for the nth term, $T_n$. Since the first differences are constant (5), the sequence is linear: $T_n = an + b$. Here, $a = 5$, so $T_n = 5n + b$. Using $T_1 = 7$ (when n=1, $T_n=7$): $7 = 5(1) + b$ $7 = 5 + b$ $b = 2$ So, the nth term is $T_n = 5n + 2$. Now, find the 10th term ($T_{10}$): $T_{10} = 5(10) + 2 = 50 + 2 = 52$.
Question 3: Determine the formula for the nth term ($T_n$) of the sequence: 1, 8, 27, 64, ...
Solution: List terms and positions: n: 1, 2, 3, 4 $T_n$: 1, 8, 27, 64 Observe the relationship between 'n' and $T_n$: $T_1 = 1 = 1^3$ $T_2 = 8 = 2^3$ $T_3 = 27 = 3^3$ $T_4 = 64 = 4^3$ The pattern is that each term is the cube of its position number.
Therefore, the formula for the nth term is $T_n = n^3$.
Question 4: A builder stacking concrete blocks for a foundation observes the following pattern in the number of blocks in successive layers (from top to bottom): 4, 7, 12, 19, 28, ... Find the formula for the nth term ($T_n$) of this sequence.
Solution: List terms and positions: n: 1, 2, 3, 4, 5 $T_n$: 4, 7, 12, 19, 28 Calculate first differences: $7 - 4 = 3$ $12 - 7 = 5$ $19 - 12 = 7$ $28 - 19 = 9$ (First differences: 3, 5, 7, 9 - not constant)
Calculate second differences: $5 - 3 = 2$ $7 - 5 = 2$ $9 - 7 = 2$ (Second differences: 2, 2, 2 - constant) Since the second differences are constant, the sequence is quadratic, $T_n = an^2 + bn + c$. $2a = 2 \Rightarrow a = 1$. So, $T_n = n^2 + bn + c$. Using $T_1 = 4$ (when n=1, $T_n=4$): $4 = (1)^2 + b(1) + c \Rightarrow 4 = 1 + b + c \Rightarrow b + c = 3$ (Eq 1) Using $T_2 = 7$ (when n=2, $T_n=7$): $7 = (2)^2 + b(2) + c \Rightarrow 7 = 4 + 2b + c \Rightarrow 2b + c = 3$ (Eq 2)
Solve the system: (Eq 2) - (Eq 1): $(2b + c) - (b + c) = 3 - 3 \Rightarrow b = 0$. Substitute $b=0$ into Eq 1: $0 + c = 3 \Rightarrow c = 3$.
Therefore, the formula for the nth term is $T_n = n^2 + 0n + 3$, which simplifies to $T_n = n^2 + 3$.
Verification: $T_1 = 1^2 + 3 = 4$ $T_2 = 2^2 + 3 = 7$ $T_3 = 3^2 + 3 = 12$ The formula is correct.
Example 1: Finding the nth term of a linear sequence.
A farmer in Kano observes that the number of yam seedlings planted in successive rows follows a pattern: 5, 8, 11, 14, ...
Find the formula for the nth term, $T_n$.
Solution:
List terms and positions:
n: 1, 2, 3, 4
$T_n$: 5, 8, 11, 14
Calculate first differences:
$8 - 5 = 3$
$11 - 8 = 3$
$14 - 11 = 3$
The first differences are constant (3). This indicates a linear sequence of the form $T_n = an + b$.
Here, $a = 3$. So, $T_n = 3n + b$.
Substitute the value of $T_1$ (n=1, $T_n$=5) into the formula to find 'b':
$5 = 3(1) + b$
$5 = 3 + b$
$b = 5 - 3 = 2$
Therefore, the formula for the nth term is $T_n = 3n + 2$.
Verification:
$T_1 = 3(1) + 2 = 5$
$T_2 = 3(2) + 2 = 8$
$T_3 = 3(3) + 2 = 11$
The formula is correct.
Example 2: Finding the nth term of a quadratic sequence.
A sequence is given as: 2, 6, 12, 20, 30, ...
Find the formula for the nth term, $T_n$.
Solution:
List terms and positions:
n: 1, 2, 3, 4, 5
$T_n$: 2, 6, 12, 20, 30
Calculate first differences:
$6 - 2 = 4$
$12 - 6 = 6$
$20 - 12 = 8$
$30 - 20 = 10$
(First differences: 4, 6, 8, 10 - not constant)
Calculate second differences:
$6 - 4 = 2$
$8 - 6 = 2$
$10 - 8 = 2$
(Second differences: 2, 2, 2 - constant)
Since the second differences are constant, the sequence is quadratic, with the form $T_n = an^2 + bn + c$.
The constant second difference is $2a$. So, $2a = 2 \Rightarrow a = 1$.
Substitute $a=1$ into the general form. Now $T_n = n^2 + bn + c$.
Use $T_1$ (n=1, $T_n$=2):
$2 = (1)^2 + b(1) + c$
$2 = 1 + b + c \Rightarrow b + c = 1$ (Equation 1)
Use $T_2$ (n=2, $T_n$=6):
$6 = (2)^2 + b(2) + c$
$6 = 4 + 2b + c \Rightarrow 2b + c = 2$ (Equation 2)
Solve the system of equations for b and c:
(Equation 2) - (Equation 1):
$(2b + c) - (b + c) = 2 - 1$
$b = 1$
Substitute $b=1$ into Equation 1:
$1 + c = 1 \Rightarrow c = 0$
Therefore, the formula for the nth term is $T_n = 1n^2 + 1n + 0$, which simplifies to $T_n = n^2 + n$.
Verification:
$T_1 = 1^2 + 1 = 2$
$T_2 = 2^2 + 2 = 6$
$T_3 = 3^2 + 3 = 12$
The formula is correct.
Financial Planning and Savings (Community/Economy): Application: Teachers can use examples of market women saving a fixed amount daily or weekly in a susu (thrift collection) scheme. If a woman saves N500 on day 1, N550 on day 2, N600 on day 3, etc., this forms an arithmetic sequence. Understanding the nth term helps her predict how much she will have saved on any given day or week, or how long it will take to reach a target amount.
Integration: Students can be asked to model a simple susu contribution plan for a period of weeks and determine the total savings. Population Growth Models (Environment/Community): Application: While actual population growth is complex, simplified models can be introduced using sequences. For example, if a village's population increases by a constant number of people each year (e.g., 50 people annually), this forms an arithmetic sequence. If it increases by a percentage (e.g., 2% annually), it forms a geometric sequence. This helps in understanding resource planning and infrastructure needs.
Integration: Students can be given data for a small community's population over a few years (e.g., 2000, 2050, 2100, 2150...) and asked to find the formula to predict the population in a future year. Architectural Patterns and Craft Design (Culture/Environment): Application: Many traditional Nigerian designs, like those found in Ankara fabrics, Aso-Oke weaving, or intricate carvings, feature repetitive and sequential patterns. The number of specific motifs in a row, the increasing/decreasing size of elements, or the arrangement of colors often follows a mathematical sequence.
Integration: Students can be shown images of Nigerian craftwork and asked to identify numerical patterns within the designs, perhaps counting specific elements in successive rows or sections to derive a sequence.