QUARTER 1 Module 1: Generating Patterns
● Pattern - it is formed if a set of shapes, numbers, or designs is repeated.
● Term - each number in the sequence
● Sequence - 1. a set of numbers arranged in a definite order. 2. An ordered
set of elements that can be put into one-to-one correspondence with the
set of positive integers.
Each member or element in the sequence is called a term ( a mathematical
expression that forms part of a fraction or proportion,is part of a series, or is
associated with another by a plus or minus sign.)
The term in a sequence can be written as 𝑎1, 𝑎2, 𝑎3, 𝑎4, … , 𝑎𝑛, …, which means 𝑎1 is
the first term, 𝑎2 is the second term, 𝑎3 is the third term, … , 𝑎𝑛 is the nth term, and
so on.
Sequence are classified as finite and infinite:
1. Finite sequence - having a countable number of elements. This means it
has an end or last term.
Examples: a) Days of the week (sunday, monday, tuesday, wednesday, … ,
saturday)
b) Alphabet ( a, b, c, d, e, f, g, … , x, y, z)
2. Infinite sequence - extending indefinitely or having unlimited spatial
extent. The number of terms of the sequence continues without stopping
or it has no end term. The ellipsis (...) implies that the numbers continue
forever.
Examples: a) counting numbers ( 1, 2, 3, 4, 5, ….)
b) the positive even numbers (2, 4, 6, 8, 210, … )
There are some special sequences that you should recognise.
The most important of these are:
2
● Square numbers: 1, 4, 9, 16, 25, 36, ... - the nth term is 𝑛
3
● Cube numbers: 1, 8, 27, 64, 125, ... - the nth term is 𝑛
● Triangular numbers: 1, 3, 6, 10, 15, ... (these numbers can be represented as
a triangle of dots). The term to term rule for the triangle numbers is to add
one more each time: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 etc.
● Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ... (in this sequence you start off with 1
and then to get each term you add the two terms that come before it)
,Sometimes a pattern in the sequence can be obtained and the sequence can be
2 3 4 5 6
written using a general term. In the previous example, x, 2𝑥 , 3𝑥 , 4𝑥 , 5𝑥 , 6𝑥 , … ,
each term has the same exponent and coefficient.
𝑛
We can write this sequence as 𝑎𝑛 =𝑛𝑥 where n=1, 2 ,3, 4, 5, 6, …, 𝑎𝑛 is called the
general or nth term.
A. Finding several terms of a sequence, given the general term:
Example 1.
Find the first four terms of the sequence 𝑎𝑛 = 2𝑛 − 1
Solution: To find the first term, let n= 1
𝑎𝑛 = 2𝑛 − 1 use the given general term
𝑎1 = 2(1) − 1 substitute n by 1
𝑎1 = 2 − 1 perform the operations
𝑎1 = 1 simplify
Find the second term, 𝑛 = 2
𝑎2 = 2(2) − 1
𝑎2 = 4 − 1
𝑎2 = 3
Find the third term, 𝑛 = 3
𝑎3 = 2(3) − 1
𝑎3 = 6 − 1
𝑎3 = 5
Find the fourth term, 𝑛 = 4
𝑎4 = 2(4) − 1
𝑎4 = 8 − 1
𝑎4 = 7
Therefore, the first four terms of the sequence are 1, 3, 5, 7
Example 2.
𝑛
(−1)
Find the 5th terms of the sequence 𝑏 = 𝑛+1
𝑛
Solution: To find the 5th term, let n= 5
5
(−1)
𝑏5 = 𝑛+1
use the given general term
, 5
(−1)
𝑏5 = 5+1
substitute n by 5
−1 1
𝑏5 = 6
=- 6
simplify (-1 raised to an odd number power is always negative)
B. Finding the general term, given several terms of the sequence:
Example 3.
Write the general term of the sequence 5, 12, 19, 26, 33,..
Solution: Notice that each term is 7 more than the previous term. We can
search the pattern using a tabular form.
In the pattern, the number of times that 7 is added to 5 is one less than the nth
term (n – 1). Thus,
𝑎𝑛 = 5 + 7(𝑛 − 1) equate 𝑎 𝑎𝑛𝑑 5 + 7(𝑛 − 1)
𝑛
𝑎𝑛 = 5 + 7𝑛 − 7) apply distributive property of multiplication
𝑎𝑛 = 7𝑛 − 2 combine similar terms
Therefore, the nth term of the sequence is 𝑎 = 7𝑛 − 2, where n= 1,2,3,4,5, …
𝑛
Example 4.
Write the general term of the sequence 2, 4, 8, 16, 32, . ..
Solution: Notice that each term is 2 times the previous term. We can search the
pattern using a tabular form.
𝑛
Therefore, the nth term of the sequence is 𝑎 = 2 , where n=1,2,3,4,5,...,
𝑛
, Example 5.
1 1 1 1
Find the general term of the sequence 1, 4 , 9 , 16 , 25 , ...
1 1 1 1 1 1
Solution: ,
1 4
, 9
, 16
, 25
, ... write 1 as 1
1 1 1 1 1 1
2 , 2 , 2 , 2 ' 2 , ..., 2 notice each denominator is an integer
1 2 3 4 5 𝑛
squared
1
Therefore, the nth term of the sequence is 𝑎 = 2 , where n=1,2,3,4,5,...,
𝑛 𝑛
● Pattern - it is formed if a set of shapes, numbers, or designs is repeated.
● Term - each number in the sequence
● Sequence - 1. a set of numbers arranged in a definite order. 2. An ordered
set of elements that can be put into one-to-one correspondence with the
set of positive integers.
Each member or element in the sequence is called a term ( a mathematical
expression that forms part of a fraction or proportion,is part of a series, or is
associated with another by a plus or minus sign.)
The term in a sequence can be written as 𝑎1, 𝑎2, 𝑎3, 𝑎4, … , 𝑎𝑛, …, which means 𝑎1 is
the first term, 𝑎2 is the second term, 𝑎3 is the third term, … , 𝑎𝑛 is the nth term, and
so on.
Sequence are classified as finite and infinite:
1. Finite sequence - having a countable number of elements. This means it
has an end or last term.
Examples: a) Days of the week (sunday, monday, tuesday, wednesday, … ,
saturday)
b) Alphabet ( a, b, c, d, e, f, g, … , x, y, z)
2. Infinite sequence - extending indefinitely or having unlimited spatial
extent. The number of terms of the sequence continues without stopping
or it has no end term. The ellipsis (...) implies that the numbers continue
forever.
Examples: a) counting numbers ( 1, 2, 3, 4, 5, ….)
b) the positive even numbers (2, 4, 6, 8, 210, … )
There are some special sequences that you should recognise.
The most important of these are:
2
● Square numbers: 1, 4, 9, 16, 25, 36, ... - the nth term is 𝑛
3
● Cube numbers: 1, 8, 27, 64, 125, ... - the nth term is 𝑛
● Triangular numbers: 1, 3, 6, 10, 15, ... (these numbers can be represented as
a triangle of dots). The term to term rule for the triangle numbers is to add
one more each time: 1 + 2 = 3, 3 + 3 = 6, 6 + 4 = 10 etc.
● Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ... (in this sequence you start off with 1
and then to get each term you add the two terms that come before it)
,Sometimes a pattern in the sequence can be obtained and the sequence can be
2 3 4 5 6
written using a general term. In the previous example, x, 2𝑥 , 3𝑥 , 4𝑥 , 5𝑥 , 6𝑥 , … ,
each term has the same exponent and coefficient.
𝑛
We can write this sequence as 𝑎𝑛 =𝑛𝑥 where n=1, 2 ,3, 4, 5, 6, …, 𝑎𝑛 is called the
general or nth term.
A. Finding several terms of a sequence, given the general term:
Example 1.
Find the first four terms of the sequence 𝑎𝑛 = 2𝑛 − 1
Solution: To find the first term, let n= 1
𝑎𝑛 = 2𝑛 − 1 use the given general term
𝑎1 = 2(1) − 1 substitute n by 1
𝑎1 = 2 − 1 perform the operations
𝑎1 = 1 simplify
Find the second term, 𝑛 = 2
𝑎2 = 2(2) − 1
𝑎2 = 4 − 1
𝑎2 = 3
Find the third term, 𝑛 = 3
𝑎3 = 2(3) − 1
𝑎3 = 6 − 1
𝑎3 = 5
Find the fourth term, 𝑛 = 4
𝑎4 = 2(4) − 1
𝑎4 = 8 − 1
𝑎4 = 7
Therefore, the first four terms of the sequence are 1, 3, 5, 7
Example 2.
𝑛
(−1)
Find the 5th terms of the sequence 𝑏 = 𝑛+1
𝑛
Solution: To find the 5th term, let n= 5
5
(−1)
𝑏5 = 𝑛+1
use the given general term
, 5
(−1)
𝑏5 = 5+1
substitute n by 5
−1 1
𝑏5 = 6
=- 6
simplify (-1 raised to an odd number power is always negative)
B. Finding the general term, given several terms of the sequence:
Example 3.
Write the general term of the sequence 5, 12, 19, 26, 33,..
Solution: Notice that each term is 7 more than the previous term. We can
search the pattern using a tabular form.
In the pattern, the number of times that 7 is added to 5 is one less than the nth
term (n – 1). Thus,
𝑎𝑛 = 5 + 7(𝑛 − 1) equate 𝑎 𝑎𝑛𝑑 5 + 7(𝑛 − 1)
𝑛
𝑎𝑛 = 5 + 7𝑛 − 7) apply distributive property of multiplication
𝑎𝑛 = 7𝑛 − 2 combine similar terms
Therefore, the nth term of the sequence is 𝑎 = 7𝑛 − 2, where n= 1,2,3,4,5, …
𝑛
Example 4.
Write the general term of the sequence 2, 4, 8, 16, 32, . ..
Solution: Notice that each term is 2 times the previous term. We can search the
pattern using a tabular form.
𝑛
Therefore, the nth term of the sequence is 𝑎 = 2 , where n=1,2,3,4,5,...,
𝑛
, Example 5.
1 1 1 1
Find the general term of the sequence 1, 4 , 9 , 16 , 25 , ...
1 1 1 1 1 1
Solution: ,
1 4
, 9
, 16
, 25
, ... write 1 as 1
1 1 1 1 1 1
2 , 2 , 2 , 2 ' 2 , ..., 2 notice each denominator is an integer
1 2 3 4 5 𝑛
squared
1
Therefore, the nth term of the sequence is 𝑎 = 2 , where n=1,2,3,4,5,...,
𝑛 𝑛
