SM015 - 3.1 (Notes Solutions)

3.1 Sequences and Series

Friday, July 10, 2020 12:12 PM

Learning Outcomes:
At the end of the lesson, students should be able to
(a) Write the th term of simple sequences and series.
(b) Find the th term of arithmetic sequence and series and use the

sum formula.
(c) Find the th term of geometric sequence and series and use the

sum formula.



Type Arithmetic Geometric

[common difference, ] [common ratio, ]

term

Sum of
the first

terms

Sum to --------
infinity

In general, (applicable for any type of sequence)

term
Sum from term to

term where

3.1 Sequences and Series Page 1

Sequences & Series

Monday, June 08, 2020 7:46 AM

Sequences
 A sequence is an arrangement of a set of numbers in a particular

order followed by some rule.
 The term of a sequence is denoted as where

E.g.
 2, 4, 6, 8, 10, 12 [Finite sequence]
 3, 7, 11, 15, … [Infinite sequence]

Series where
 A series is the sum of the terms of a sequence.
 The sum of the first terms of a sequence is denoted as

E.g.
 [Finite series]
 [Infinite series]

Example 1 .
Find the term of the sequence
Solutions :

Remarks:
• The term of any sequence can be found using the formula:
.
• The sum from term to term of any sequence can be found using the formula:
.

Example
Given that a sequence with the sum of the first 4 terms is 30 and the sum of
the first 5 term is 45. Find the fifth term of the sequence.
Solutions :

3.1 Sequences and Series Page 2

Arithmetic Sequences

Monday, June 08, 2020 7:46 AM

 An arithmetic sequence is a sequence of numbers in which
each term [except the first term] can be obtained from the
previous term by adding a nonzero constant number called the
common difference.

 The arithmetic sequence with the first term, and the
common difference, is of the form

 The term of the arithmetic sequence is given by

 The common difference, is given by

 The sum of the first terms of the arithmetic sequence is
given by

OR

Remarks:

To identify Arithmetic sequence, determine whether is

independent of n or not.

[i.e. determine the existence of common difference between any two

consecutive terms]

3.1 Sequences and Series Page 3

Example 2 & 8

Friday, May 29, 2020 10:48 PM

Example 2

The term of an arithmetic sequence is 52 and the term is 92.

Find the term

Solution:

Example 8

A lecture hall has 50 rows of seats with 30 seats in the first row,
32 in the second, and 34 in the third and so on. Find the total number
of seats.
Solution:

Let the number of seats in the row
Total seat

There are total of 3950 seats.

3.1 Sequences and Series Page 4

Example 3

Friday, May 29, 2020 10:48 PM

In an arithmetic sequence, the term is 3 and the sum for the first
six terms is 76.5. Find
(a) the first term and the common difference,
Solution:

(b) the number of terms to be taken so that its sum is zero.
Solution:

Since is a positive integer, [can divide both sides with ]

after divide both sides with

23 terms are taken.

3.1 Sequences and Series Page 5

Geometric Sequences

Monday, June 08, 2020 7:46 AM

 An geometric sequence is a sequence of numbers in which
each term [except the first term] can be obtained from the
previous term by multiply by a nonzero constant number [ ]
called the common ratio.

 The geometric sequence with the first term, and the
common ratio, is of the form

 The term of the geometric sequence is given by

 The common ratio, is given by

 The sum of the first terms of the geometric sequence is
given by

 The sum to infinity of the geometric sequence is given by

Remarks:
To identify geometric sequence, determine whether is
independent of n or not.
[i.e. determine the existence of common ratio between any two
consecutive terms]

3.1 Sequences and Series Page 6

Example 4 . Find the

Friday, May 29, 2020 10:48 PM

Given the sequence

(a) term
Solution:

(b) term
Solution:

If the term is 192, find the value of k.
Solution:

3.1 Sequences and Series Page 7

Example 5

Friday, May 29, 2020 10:48 PM

The first term of a geometric sequence is 27 and the common ratio is .
(a) Find the sum of the first 5 terms.
Solution:

(b) Find the least number of terms the sequence can have if the sum
exceeds 550.

Solution:

The least number of terms required is 8.

3.1 Sequences and Series Page 8

Example 6 & 7 .

Friday, May 29, 2020 10:48 PM

Example 6

Find the sum to infinity for the series

Solution:

Example 7

Express as an infinite geometry series and stated as a fraction in
the simplest form.

Solution:

3.1 Sequences and Series Page 9

Derivation for in Arithmetic Sequence

Friday, July 30, 2021 10:39 PM

Consider the arithmetic sequence with the first term, and the common
difference, such that

Note that

The term of an
arithmetic sequence is
given by

Note that ------(1)
Also
(1)+(2) [By commutative law in addition]

------(2)

The sum of the first terms of an
arithmetic sequence is given by

3.1 Sequences and Series Page 10

Derivation for in Geometric Sequence

Friday, July 30, 2021 10:39 PM

Consider the geometric sequence with the first term, and the
common ratio, such that

Note that

The term of a geometric sequence
is given by

Note that ------(1)
If multiply (1) with , ------(2)

The sum of the first terms of a
geometric sequence is given by

Note that as for and therefore

The sum to infinity of a
geometric sequence is given by

for

3.1 Sequences and Series Page 11