Successful Minds,Making Mathematics number patterns &sequences Simple.

the document will enrich your understanding of number patterns and sequences

1. Successful
Minds,Making
Mathematics
number patterns
&sequences
Simple.
Issue 1.
thato barry

2. 1

Successful Minds,
Making Mathematics
number patterns
&sequences
Simple.
Issue 1.










Thato Barry Mmitsa

3. 2






TABLE OF CONTENT.

Paragraphs Page
Paragraph 1.1 : Number patterns and Sequences 4 - 6
Paragraph 1.1.1: quadratic number pattern 7 -11
Paragraph 2.1 : Arithmetic Sequences 12 - 15
Paragraph 2.1.1: Geometric Sequences. 16 - 20

4. 3

Introductory Remarks.
How to use the book :
 The book is meant to be supplementary to your class work and textbook
 The book gives you the opportunity to do extra exercises on Number patterns
and sequences.
 The exercises help you to revise and prepare well for tests.























The establishment of this book, was after great influence of mathematics especially
number patterns from Elna van Krandenburg, Nelis Vermuelen and Elise Prins the
authors of the mathematics guide ( Pythagoras 10-12 CAPS-KABV)

5. 4


PARAGRAPH 1.1 NUMBER PATTERNS AND SEQUENCES.
A pattern is an arrangement of things using a rule or rules in mathematics. Patterns
can be found in nature, shapes, events and sets of numbers .A Number pattern is a
list of numbers that follows a certain pattern or sequence. For example:
 1; 3; 5; 7 is an odd sequences.
 2; 4; 6; 8 is an even sequences
Every number in the row is called a term. The value of the term is indicated by the
letter T. For example in the above sequences �
2=3 ��� 4
�
1=�=1
�
� Is the value of the �
�ℎ
term, �
� is also called the general term. The general term is
algemene also gives the rule by which each term is term formed, e.g. for the row of
odd numbers:
 �
�=2�−1 where n € ₭
 ??????���??????�� ��??????�??????�� �
�=��+�
To determine �
� we can use inspection or a table.
 By inspection
e.g. for the sequence 3; 5; 7; 9….. The common difference between
consecutive terms is d= 2. Look for the pattern
�
1=1∗2+1
�
2=2∗2+1
 Using a table.
Consider the sequence 2; 5; 8; 11…..
Position of the term 1 2 3 4
Value of the term 2 5 8 11


The difference between the values of consecutive terms is d=3. Therefore �
�
has the form,3�+�, this means that:
�
�=3∗1+� But �
1=2 therefore substitute

→ 2=3+� →�= −1
General term �
�=3�−1

6. 5






Example 1.
Consider the following pattern 1; 4; 7;…
a) Write down the following three terms.
b) Write down the different consecutive term.
SOLUTIONS
a) 1; 4; 7; 10; 13; 16
b) The difference is 3
�
2−�
1=�
3−�
2=3
Example 2.
Consider the following pattern -4; -1; 2; 5; 8
a) Write down the difference between consecutive terms
b) Copy and complete the following
�
1=1∗3− ⎕ �
2=2∗3−⎕ �
3=3∗3−⎕ �
4=4∗3−⎕ �
5=5∗3−⎕
c) What is the value of the 50
th
term in the number pattern?
d) Find a formula�
�, the general term of the number pattern.
SOLUTIONS
a) �
2−�
1= −1−(−4)=�
b) �
1=1∗3− 7=−4 �
2=2∗3−7=−1 �
3=3∗3−7=2 �
4=4∗3−7=5
�
5=5∗3−7=8
c) �
50=50∗3−7=143
d) �
�=3�−7
Example 3
Consider the following pattern 1; -3; -7; -11; …..
a) What is the difference between consecutive terms?
b) Find the formula for �
�, the value of the nth term.
SOLUTIONS.
a) �
2−�
1=−3−1=−4
b) �
�=��+�
�
2=(−4)�+�
−3=(−4)(2)+�
�=5
�
�=−4�+5

7. 6



Practice Exercise A.
1) Consider the following patterns.
a) -1; 4; 9; 14; ……
b) 800; 400; 200; 100
Write down the next 3 numbers in each sequence.
2) Consider the following pattern 1; 1.5; 2; 2.5; ….
a) Write down the difference between the consecutive terms.
b) Copy and complete the following
�
1=1∗0.5+⎕; �
2=2∗0.5+⎕; �
3=3∗0.5+⎕; �
4=4∗0.5+⎕;

3) The following sequence is given 2; 6; 10; 14; …..
a) What is the difference between the consecutive terms?
b) Write down the value of the 50
th
term.
c) Find the formula for �
�, the general term of the sequence.
d) Which term equals 118?

8. 7


Paragraph 1.1.1: Quadratic number pattern
A Quadratic is a polynomial, involving the second power (square) of a variable but no
higher powers. Quadratic numbers can be written a number sequence, e.g. the even
numbers.
 2; 4; 6; ….
Each number is still called a term. E.g .�
1=2, the first term. �
� Is still the general term
or �
�ℎ
term of a sequence. Thus the formula �
�=��+� describes the rule which
each term in a row is determined.
Example 1
Consider the following pattern 2; 4; 6; …..
Find the 10
th
term.
SOLUTION
d=2
�
�=2�
�
10=2(10)=20
If the first differences are constant, the formula for the general term is of the first
degree. i.e linear:
 E.g. 3; 5; 7; 9; 11; ………
 First differences 2; 2; 2; 2
 General term �
�=2�+1

If the second differences are constant, the formula for the general term is of the
second degree. i.e. quadratic:
 2; 5; 10; 17; 26; ……….
 First differences: 3; 5; 7; 9;
 Second differences: 2; 2; 2; 2
 General term: �
�=�
2
+1 .

9. 8

To determine the General term (�
�), we can use inspection or a table to determine
the relationship between the position of a term (n) and its value (�
�). i.e. find the
function rule using a tabular form.
Consider the following pattern -1; 5; 15; 29; 47; ….
Position of term (n) 1 2 3 4 5
Value of the term (�
�) -1 5 15 29 47

 The first differences: 6; 10; 14;18
 The second differences: 4; 4; 4;
Therefore the second differences are constant, thus the formula for quadratic
�
�=��
2
+��+�
 To find a: �=
������ �??????���������
2
=
4
2
=2
 So �
�=2�
2
+��+�
To find b and c:
Substitute any two numbers paired into �
�, and solve the two equations
simultaneously.
E.g. (1; -1) equation 1
 −1=2(1)
2
+�(1)+�
 −3=�+�
(2; 5) equation 2
 5=2(2)
2
+�(2)+�
 −3=2�+�
Equation 1 ─ Equation 2
−�=0
�=0
Substitute b=0 into Equation 1
 −3=0+�
 �=−3
Therefore �
�=2(�)
2
−3
Learn this.
Consider the following pattern 3; 6; 11; 18; …
 First differences: 3; 5; 7
 Second differences: 2; 2;
 General term: �
�=�
2
+2
NB!!!!!! Learn this.
Mathematics uses a rule to solve equation 1, 2 and 3 simultaneously.

10. 9

 �+�+�=3 equation 1 using only the first term in the row.
 3�+�=3 equation 2 using only the first term in the first
differences.
 2�=2 equation 3 using only the first term in the second
differences.s
Thus:


 2�=2
a=1
 3�+�=3=3(1)+�
b=0
 a+ b+ c=3=1+0+c
 c=2
�
�=��
2
+��+�
�
�=(1)�
2
+0(�)+2
�
�=�
2
+2


Example 1.
Consider the following pattern: -6: 0; 8; 18; ….
a) Write down the next two terms.
b) Calculate the �
�ℎ
term or general term of the pattern.
c) What is the value of the 30
th
term?


SOLUTIONS.
a) The first difference is 6; 8: 10
The second difference is 2; 2:
Note that the first difference is determined by adding 2 the second difference, so the
next first difference is 12. We add the difference 12 to the number before the term to
determine the term = 30
First row: -6; 0; 8; 18; 30; 44; …
First differences: 6; 8;10; 12; 14
Second differences: 2; 2; 2; 2;
b) 2a=2
a=1
3a+b=6=3(1)+b
b=3
a+ b+ c= -6= 1+3+c= -6
c= -2
therefore �
�=��
2
+��+�=(1)�
2
+(3)�−2
�
�=�
2
+3�−2
c) �
30=(30)
2
+3(30)−2.

11. 10

�
�=988

Example 2.
Consider the following pattern 2; 8; 16; 26; …..
a) Determine the value of the 6
th
and 7
th
term.
b) Calculate the �
�ℎ
, or general term of the pattern.
c) What is the value of �
67?
d) In what position is term 178?

SOLUTIONS.
a) ... 16; 26; …; 52; 58.
Find the missing number in between using example 1.
b) 2a=2
a=1
3a+b=6=3(1) +b
b=3
a+ b+ c=2= 1+3
c= -2
�
�=��
2
+��+�=(1)�
2
+(3)�−2
�
�=�
2
+3�−2
c)
�
67=(67)
2
+3(67)−2
�
67=4688
d) 178=�
2
+3�−2
�
2
+3�−180=0
(�+15)(�−12)=0
�≠−15 �?????? �=12

NB!!!!!!!!! n is a positive integer

12. 11






Practice Exercise B.


1) For each number pattern.
a) Write down the next three terms.
b) Find the formula of the general term.
c) Use the general term to find �
9 in:

(i) 1; 5; 11; 19; 29; …..
(ii) 2; 4; 8; 16; 26; …
(iii) 4; 8; 16; 28; 44; …

2) The pattern 3; 8; x; 24; …… is a quadratic pattern
a) Calculate x.
b) Determine the �
�ℎ
term of the pattern.
c) Which term has a value just smaller than 200?

13. 12




Paragraph 2.1 : Arithmetic Sequences.
Arithmetic or Arithmetics is from the Greek word, meaning the study of numbers. In
a sequence which is Arithmetic, there is a constant difference (d) between the
consecutive terms. E.g. 2; 4; 6; … the difference d=2.
Therefore d=??????
??????−??????
??????−�, and the sequence �
1; �
2; �
3;… is an arithmetic sequence if:
�
2−�
1=�
3−�
2 .
The first term is usually indicated by the symbol a, meaning that the general term of
an arithmetic sequence is ??????
??????=??????+(??????−�)??????
Consider the following Arithmetic sequence: 5; 2; -1; -4; …
d=-3 �
�=5−(�−1)(−3)=5−3�+3
�
�=−3�+8
NOTE: since the first difference in an Arithmetic sequence remains constant, �
� will
be given by a linear expression. �
� Can also be determined by using the function
approach, as previously.
 The sum of n terms of an Arithmetic sequence is:
�
�=
�
2
[2�+(�−1)�] �?????? �
�=
�
2
[�+�]
Where l, is the last term in the Arithmetic sequence
Consider the above Arithmetic sequence 5; ……;-4; ….the sum of the first 10 terms
is: �
10=
10
2
[2(5)+(10−1)(−3)]
�
10=−85

Example 1

Consider the following sequences, give the next three terms of each row.
a) 3; 7; 11; 15; …
b) -4; 6; 11; 16; …


SOLUTIONS.

14. 13

a) �=�
2−�
1=7−3=11−7=4
So we add the consecutive difference to the previous term to find the next
term.
19; 23; 27 are the next three terms.
b) d=5
21; 26; 31 are the next three terms.


Example 2
Consider the following Arithmetic sequences, find the general term in each case and
determine the sum of 20 terms.
a) 2; 4; 6; 8; …...
b) 9; 6; 3; 0 …….



SOLUTIONS.
a) a= 2 ??????=4−2=6−4=2
�
�=2+(�−1)(2)
�
�=2�
b)
c)
d) a= 9 ??????=6−9=3−6=−3-------------
�
�=9+(�−1)(−3)
�
�= −3�+6

15. 14



Practice Exercise C.

1) For each Arithmetic sequence determine the:
a) Next three terms of the row.
b) The general term of the sequence.
c) Find term number 20.
(i) 14; 13; 12; 11
(ii) 301; 298; 295, 292; ….

2) Given the Arithmetic sequence 7; 2; -3; …determine.
a) The general term �
�.
b) The 9
th
term.
c) The sum of the first 20 terms.
3) The first three terms of an Arithmetic sequence are:
�??????+�;�??????;�??????−??????
Calculate the second term.


TRY OUT THE QUIZZES ONLINE

 https://za.ixl.com/math/grade-10/geometric-sequences
 http://www.quiz-magic.com/m/q.aspx?t=368&d=36



ACTIVITY 1.1; you now have 2 tasks to complete
 Task 1 on paragraph 1.1
 Task 2 on paragraph 1.1 and 1.1.1

Rubric for Assessing activity task 1& 2
Score Description



9-10
The assesse has demonstrated a full and complete understanding of the
mathematical content and practices essential to this task. The student has
addressed the task in a mathematically sound manner. The response contains
evidence of the student’s competence in problem solving, reasoning, and/or
modelling to the full extent that these processes apply to the specified task. The
response may, however, contain minor flaws that do not detract from a
demonstration of full understanding.

16. 15




7-8
The assesse has demonstrated a reasonable understanding of the
mathematical content and practices essential to this task. The student has
addressed most of the task in a mathematically sound manner. The response
contains sufficient evidence of the student’s competence in problem solving,
reasoning, and/or modelling, but not enough evidence to demonstrate a full
understanding of the processes he or she applies to the specified task. The
response may contain errors that can be attributed to misinterpretation of the
prompt; errors attributed to insufficient, non-mathematical knowledge; and
errors attributed to careless execution of mathematical processes


5-6

The assesse has demonstrated a partial understanding of the mathematical
content and practices essential to this task. The student’s response contains
some of the attributes of an appropriate response but lacks convincing evidence
that the student fully comprehends the essential mathematical ideas addressed
by this task. Such deficits include evidence of insufficient mathematical
knowledge; errors in fundamental mathematical procedures; and other
omissions or irregularities that bring into question the student’s competence in
problem solving, reasoning, and/or modelling as applied to the specified task.


3-4
The assesse has demonstrated a limited understanding of the mathematical
content and practices essential to this task. The student’s response is
incomplete and exhibits many errors. Although the student’s response has
addressed at least one of the conditions of the task, the student reached an
inadequate conclusion and/or demonstrated problem solving, reasoning, and/or
modelling that was faulty or incomplete as related to the specified task.

1-2 The assesse has demonstrated merely an acquaintance with the topic, or
provided a completely incorrect or uninterpretable response. The student’s
response may be associated with the task, but contains few attributes of an
appropriate response. There are significant omissions or irregularities that
indicate a lack of comprehension in regard to the mathematical content and
practices essential to this task. No evidence is present that demonstrates the
student’s competence in problem solving, reasoning, and/or modelling related to
the specified task.
0 The assesse did not attempt to do the task.


17. 16


Paragraph 2.1.1: Geometric Sequences.
In mathematics a geometric sequence, is a sequence of numbers where each term
after the first is found by multiplying the previous one by a fixed, non-zero number
called the common ration. In a geometric sequence there is always a constant ration
(r) between the consecutive terms. E.g. 2; 4; 8; 16 (r = 2)
1;−
1
2
;
1
4
−
1
8

Therefore, ??????=
??????
??????
??????
??????−1

And
??????
??????
??????
??????−1
=
??????
??????+1
??????
??????
=
??????
2
??????
1
=
??????
3
??????
2

Thus the general term of a Geometric Sequence is
�
�=�.??????
�−1


E.g. in the above geometric sequence:
r = 2 �
�=(2)(2)
�−1
=2∗2
�
∗2
−1

�
�=2
�

The sum of n terms of a Geometric sequence is �
�=
??????(1−�
??????
)
1−�
, usually when r < 1.
Or &#3627408454;
&#3627408475;=
??????(&#3627408479;
??????
−1)
&#3627408479;−1
, usually when r > 1
E.g. the above geometric sequence, find the sum of the first ten terms.
&#3627408454;
10
2(2
10
−1)
2−1
=2046
 Note that the sum of a sequence is known as a series
a) 2; 4; 8; 16 is a sequence
b) 2 + 4 + 8+ 16 is a series.
WE also have the Sum to infinity determined by: &#3627408454;
∞=
??????
1−&#3627408479;
, when -1< r < 1 thus the
sequence is convergent.

Example 1.
Consider the following Geometric sequence: 1;3; 9;
a) Find the next three terms.
b) Find the general term &#3627408455;
&#3627408475;.

18. 17

c) Find the sum of 10 terms.


SOLUTIONS
a) r= -3=
??????
2
??????
1
=
??????
3
??????
2
=
−3
1
=
9
−3

next term -3*9= -27
-3*-27= 81
-3*81= -243
1; 3; 9; -27; 81; -243
b) &#3627408455;
&#3627408475;=&#3627408462;.??????
&#3627408475;−1
=(1)(−3)
&#3627408475;−1

&#3627408455;
&#3627408475;=(−3)
&#3627408475;−1

c) &#3627408454;
10=
(1)(−3
10
−1)
−3−4
=14762

Example 2
The first term and common ration of a geometric sequence are 2 and
3
2
respectively.
a) Determine the value of the third (3
rd
) term of the series.
b) Show that the sum to n terms of the series is &#3627408454;
&#3627408475;=4[(
3
2
)
&#3627408475;
−1)
c) Hence, calculate n for which &#3627408454;
&#3627408475;=
65
4




SOLUTIONS.
a) &#3627408455;
&#3627408475;=(2)(
3
2
)
&#3627408475;−1

. &#3627408455;
3=(2)(
3
2
)
3−1
=
9
2

b) &#3627408454;
&#3627408475;=
(2)[(
3
2
)
??????
−1)
3
2
−1
=
(2)[(
3
2
)
??????
−1)
1
2

. &#3627408454;
&#3627408475;=4[(
3
2
)
&#3627408475;
−1)
c)
65
4
=4[(
3
2
)
&#3627408475;
−1)
. 65=16[(
3
2
)
&#3627408475;
−1)

19. 18

.
65
16
=(
3
2
)
&#3627408475;
−1
. (
3
2
)
&#3627408475;
=
81
16

. (
3
2
)
&#3627408475;
=(
3
2
)
4

. n= 4




Practice Exercise D


1) Consider the following geometric sequence:
45
4
+
135
16
+
405
64

a) Find the next three terms.
b) Find the general term &#3627408455;
&#3627408475;.
c) Find the sum of 10 terms.
2) The first and the sixth term of a geometric sequence series are equal to 8 and
1
4
respectively. Use formulas and determine without the use of a calculator.
a) The common ration (r)
b) The sum of the first six terms of the series.





Activity 1.2:
Now that you reached the end of the book you have to complete the following
tasks:
 Task 4
 Task 5

Rubric for assessment task 4 & 5

20. 19

Score 0-2 3-5 6-8 9-10 Total
Problem solving No strategy is
chosen or a
strategy is
chosen that
will not lead to
a solution.
Little or no
evidence of
engagement in
the task is
present.
A partially
correct strategy
is chosen, or a
correct strategy
for only solving
part of the task
is chosen.
Evidence on
some relevant
previous
knowledge is
present
showing
relevant
engagement
with the task
A correct
strategy is
chosen based
on the
mathematical
situation in
the task.
Evidence of
solidifying
prior
knowledge
and applying
it to the
problem
solving
situation is
present.
An efficient
strategy is
chosen and
progress
towards the
solution is
evaluated.
Evidence of
analysing the
situation in
mathematical
terms and
extending
prior
knowledge is
present.

Reasoning and
proof
Arguments are
made with no
mathematical
basis.
No correct
reasoning nor
justification for
reasoning is
present.
Arguments are
made with
some
mathematical
bases.
Some correct
reasoning or
justification for
reasoning is
present.
Arguments
are
constructed
with adequate
mathematical
bases.
A systematic
approach or
justification of
correct
reasoning is
present
Deductive
arguments
are used to
justify
decisions and
may result in
formal proof.
Evidence is
used to justify
and support
decisions
made and
conclusions
reached

Connections No formal
mathematical
terms or
symbolic
notations are
evident.
An attempt is
made to use a
formal math
language. One
formal math
term or
symbolic
notation is
present.
Formal math
language is
used for
clarifying
ideas. At least
two formal
terms or
symbolic
notations are
evident.
Formal math
language is
used and
symbolic
notations
beyond grade
level.

Representations No attempt is
made to
construct a
mathematical
representation.
An attempt is
made in
representations
but problem
solving is not
accurate
An
appropriate
and accurate
representation
is constructed
and refined to
solve
problems.
An
appropriate
representation
is constructed
to analyse
relationships
and extend
thinking and

21. 20

clarity of
concepts.




























ACKNOWLEDGEMENTS OF THIS BOOK ;

22. 21


All rights are reserved to ELNA VAN KRANDENBURG, NELIS VERMEULEN AND
ELISE PRINS, for providing relevant information of number patterns in their grade
10-12 study guides (Pythagoras 10, 11, &12 CAPS-KABV)
And Smarter Balanced Assessment Consortium for providing Smarter balanced
General Rubrics for Mathematics.