# lesson seqs

```LESSON PLAN FOR LOWER SIXTH SCIENCE
Name of student teacher: MBURUBAH WALTERS KUWALA
Level:5
Matricule:UBa21G0558
Name of cooperative Teacher: MR DIEUDONE
School: CCAST BAMBILI
Class: LSS
Duration: 90 minutes
Term: SECOND TERM
Number on roll : Girls = 15
Boys = 12
Date: 31/01/2023
Total = 27
MODULE: 3
Topic: SEQUENCES AND SERIES
Lesson title: ARITHMETIC SEQUENCES
Lesson objectives: At the end of the lesson, the learners should be able to:
1)
2)
3)
4)
5)
Define a numeric sequence.
Calculate the nth term and common difference of an arithmetic sequence.
Solve real life problems that result in an arithmetic sequence.
Calculate the sum of the first nth terms of an arithmetic sequence
Find missing terms in a given sequence.
Prerequisite knowledge: The learners can:
1) Perform basic arithmetic operations such as addition,
subtraction, multiplication and division.
2) Calculate the average of a set of numbers
Motivation/Rationale:
ο With the knowledge of sequences, we can be able to define the things around us that occurs in patterns
ο Calculations in financial institutions involving interest rates requires the knowledge of sequences
Didactic materials:
Protractors, rulers, pencils
References: - FURTHER PURE MATHEMATICS MADE EASY, THIRD EDITION, EWANE ROLAND ALUNGE
- NEW CENTURY FURTHER PURE MATHEMATICS, AJECK LOUIS
Page 1 of 6
STAGES/
DURATION
TEACHING/LEARNING activities
Verification of prerequisite
1. Simplify the following
expressions
a.
Introduction
(15mins)
TEACHER’S
ACTIVITIES
-Write the exercise
on the chalkboard
-call out students
to response the
questions orally.
- write out the
students present on
the board.
b. ( ) ( )
2. Given
are three
consecutive numbers and
that
and
find;
a.
-poses the problem
b. Find their average in
situation orally
terms of
Problem situation
How can you help a journalist
determine the capacity of a 30
rows theater organized in such a
way that there are 20 seats in the
first row, 26 in the second row, 32
in the third row and so on in that
pattern?
LEARNERS’
ACTIVITIES
-Students try out
the exercise on
their rough
worksheets and
give response
orally when
called out by the
teacher.
- Take note of the
teacher wrote on
the board
-students listen
attentively to the
problem situation
and by a show of
hand, propose
solutions to the
problem if they
can.
LEARNING POINTS
observatio
ns
Definition of a numerical sequence
A numerical sequence is a set of numbers
occurring in a definite order. The numbers are
produced according to a well-defined rule. It
can also be considered as function with
domain and range .
A numerical sequence can be finite or infinite.
We shall use the following notations to denote
the nth term of a sequence. ,
or .
NB: the above notation is not limited only to
the above letters
Example:
1. 2, 4, 6, 8 (finite sequence)
2. 2, 4. 6. 8, . . . (infinite sequence)
Activity 1
Consider the sequence below
2, 4, 8, 16
i)
Is this sequence finite of infinite?
ii)
Identify the general term of this
sequence
TYPES OF SEQUENCES
SOLUTION TO ACTIVITY I
There are two main types of sequences which
are arithmetic and geometric sequences.
A. ARITHMETIC SEQUENCE
Lesson
Development
(65 mins)
ππ
ππ
An arithmetic sequence is a sequence whereby
the difference between any consecutive terms
is a constant called the common difference
(π).
Examples 1.
8 16 4
…
This pattern of numbers is referred to as an
Page 2 of 6
ACTIVITY II
Find the nth term of the following
arithmetic sequences below
1)
,
, …,
1
…
2) ,
,
…,
1
…
Solution
Teacher insist on
the different
domains of the two
formulae for
calculating the nth
term of an
arithmetic
sequence
Teacher moves
round the class to
make sure the
formulae a copied
correctly
1) The common
difference is = 1 and the first term
, Since n starts from
1. we substitute in the first
formula, we have
is a =
1
(
)( )
2) 1) The common
difference is = 1 and the first term
, Since n starts from 0,
we substitute in the first formula,
we have
is a =
Teacher explains
and solves one of
the questions and
leaves one for the
students.
Students carry out
the instructions,
for clarification.
arithmetic sequence. This is because the
difference between any 2 consecutive numbers
Is a constant. Here, the common difference is
8.
A.1 : The nth term of an arithmetic
sequence
The nth term of an arithmetic sequence is given
by
π»π
π»π
π)π, π
1
…
Where;
OR
Students present
their books to the
teacher for the
verification of the
formulae
π»π
π»π
ππ, π
1
…
Where;
A.2 Arithmetic mean
Students are
organized in
groups to solve
the second
question of the
activity
Let 3 numbers , b, π be and arithmetic
progression. b is called the arithmetic mean of
and π and it is defined as
π
π
π
π
A.3 Sum of the first nth terms of an
arithmetic sequence
The sum of the first nth terms of an
arithmetic sequence ( ) is given by
π
π»π
( )( )
(π
π=π
πΊπ
π
[ππ
π
(π
π)π]
0
Observe that in the above formula
1
… and thus the first term is
It is easy to see that for
1 …,
Page 3 of 6
and the formula for the first nth terms of the
arithmetic sequence beginning from zero is
given by
ACTIVITY III
π
π»π
th
Find the sum of the first n terms of
the arithmetic sequence defined by
1
1)
(
[ππ
ππ]
A.4 Sum to infinity of an arithmetic
sequence
Arithmetic sequences are divergent and so, we
cannot not sum all the terms of such a
sequence. That is the sum to infinity of an
arithmetic sequence is infinity and we write:
1
Substituting in our formula we have
)
π
π
NB: we shall assume that the domain starts
from 1 whenever it is not define.
Solution
[ (
π
π=π
…
and
πΊπ
)( )]
∞
(3+n)
π»π
1
2)
π=π
…
A.4 Relationship between the nth term and
the sum of the first nth terms of an
arithmetic sequence
Solution
and
1
The prove of this relationship will be
demonstrated in class
Substituting in our formula we have
[ ( )
(4+n)
∞
( )( )]
π»π
Writes the activity
on the board.
Moves round to
check students
work.
Guide the on the
student to go solve
on the board
πΊπ
πΊπ
π
Example
Fine an expression for the nth term of an
arithmetic sequence given that the sum of
the first nth terms is given by
4
Solution
Copy the activity
in their exercise
books. Listen
attentive to any
We know that
(
4
1)
4
Page 4 of 6
instructions and
solve in their
exercise books.
General lesson activity
Show that the sequence whose
term is given by the expression
Substituting and simplifying yields the
expression for
( )
1
…
Is an arithmetic sequence. Hence,
show that
81
(
)
=
To be solve in class
1
NB: All arithmetic sequences diverge
Resolve the problem situation
Resolve the problem
situation
Assignment
Conclusion
(10 mins)
1) Find the nth and the sum of the
first nth terms of the arithmetic
sequence defined by
15
9
1
π‘
π‘
π‘
π‘
…
4
4
2) Find an expression for the sum
of the first nth terms of the
following arithmetic sequences
a)ππ
4π
1 π
b)ππ
π
π
1
1
Teacher writes the
assignment on the
chalkboard
Students copy in
their exercise
books
…
..
Page 5 of 6
Conclusion
(10 mins)
Assignment
1) find the nth and the sum of
the first nth terms of the
arithmetic sequence defined
by
15
9
1
…
4
4
2)
.
Page 6 of 6
```