The document explains patterns and sequences in mathematics, emphasizing the importance of identifying trends among numbers to predict subsequent terms. It introduces concepts like finite and infinite sequences, general terms, and series, providing examples and rules for constructing nth terms. The text also touches on concepts of summation and the classification of sequences.

1. PATTERN
Objective:
generates patterns.
M10AL-Ia-1

3. IS THIS A PATTERN?
2,4,6,8,10,12,…
a, b, c, d, e

5. Patterns and sequences
We often need to spot a pattern in order to
predict what will happen next.
In maths, the correct name for a pattern of
numbers is called a SEQUENCE.
The first number in a SEQUENCE is sometimes
called the FIRST TERM; the second is the
SECOND TERM and so on.

6. Patterns and sequences
For any pattern it is important to try to spot
what is happening before you can predict the
next number.
The first 2 or 3 numbers is rarely enough to
show the full pattern - 4 or 5 numbers are
best.

7. Patterns and sequences
For any pattern it is important to try to spot
what is happening before you can predict the
next number.
1, 2, …… What’s the next number?

8. Patterns and sequences
For any pattern it is important to try to spot
what is happening before you can predict the
next number.
1, 2, 4,… Who thought that the next
number was 3?
What comes next?

9. Patterns and sequences
For any pattern it is important to try to spot
what is happening before you can predict the
next number.
1, 2, 4, 8, 16, …
What comes next?

10. Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5, 8, 12, 17, 23, …, …
+ 3

11. Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5, 8, 12, 17, 23, …, …
+ 3 + 4

12. Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5, 8, 12, 17, 23, …, …
+ 3 + 4

13. Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5, 8, 12, 17, 23, …, …
+ 3 + 4 + 5

14. Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5, 8, 12, 17, 23, …, …
+ 3 + 4 + 5 + 6

15. Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5, 8, 12, 17, 23, 30, …
+ 3 + 4 + 5 + 6 + 7

16. Patterns and sequences
Now try these patterns:
3, 7, 11, 15, 19, …, …
128, 64, 32, 16, 8, …, …
1000, 100, 10, 1, …, …
5, 15, 45, 135, …, …
Infinite
sequence
So what is a finite
sequence?

17. IN GENERAL,

18. CONSIDERTHE FOLLOWING SEQUENCE
• 1, 4, 9, 16, 25, …
• What is the value of a (T1 ) ?
• T5 ?
• What is the pattern?
• T1 = 1 = 12
• T2 = 4 = 22
• T3 = 9 = 32
• …
T12 = ?
• Tn = ?
Tn is
called
the
general
term
Sequence is

19. EXAMPLES
EXAMPLES: WRITE A RULE FORTHE NTH
: WRITE A RULE FORTHE NTH
TERM.
TERM.
,...
625
2
,
125
2
,
25
2
,
5
2
.
a
,...
5
2
,
5
2
,
5
2
,
5
2
4
3
2
1
,...
9
,
7
,
5
,
3
.
b
Look for a pattern…

20. EXAMPLE: WRITE A RULE FORTHE NTH
TERM.
Think:

21. Describe the pattern, write the next term, and
write a rule for the nth term of the sequence
(a) – 1, – 8, – 27, – 64, . . .
SOLUTION
You can write the terms as (– 1)3
, (– 2)3
, (– 3)3
,
(– 4)3
, . . . . The next term is a5 = (– 5)3
= – 125.
A rule for the nth term is an = (– n)3
.
a.

22. Describe the pattern, write the next term, and
write a rule for the nth term of the sequence
(b) 0, 2, 6, 12, . . . .
SOLUTION
You can write the terms as 0(1), 1(2), 2(3),
3(4), . . . .
The next term is f (5) = 4(5) = 20. A rule for the
nth term is f (n) = (n – 1)n.
b.

23. CONSIDERTHIS:
• What is the pattern? How many dots for the next term?
• What about the 50th
term?
• Need to find general term or the rule first.
…

24. Rearrange the dots:
Double the dots:

25. THUS;

26. WRITE IN GENERAL TERM
•5,8,11,14, 17, …
•25, 21, 17, 13, …
•1, 3, 9, 27

27. SERIES
• A series is the sum of the terms in the
sequence and is represented by Sn.
• E.g.
• Sn = T1 + T2 + T3 +…+ Tn
• For finite series,
• 1 + 3 + 5 + 7.
• For infinite series,
• 1 + 2 + 3 + 4 +…

28. THE SUMMATION SYMBOL

29. GENERALLY.
• For finite series
• For infinite series,

30. WRITE THE FOLLOWING SERIES USING THE
SUMMATION SYMBOL

31. FINDING THEVALUES OFTHE SUMMATION
1 +2+3+4+5+6+7+8+9+10 = 55
= 62
[2(-1) -3] + [2(0)-3]+[2(1)-3] +[2(2)-
3]
= -8

32. FINDING THE SUM FOR AN INFINITE
SEQUENCE
Infinite
sequence
Convergent
sequence
Divergent
sequence

33. CONVERGENCE

34. DIVERGENCE

35. Classify the following sequences as Finite
Sequence or Infinite sequence.
________________1. {1, 3, 5, 7, 9,…}
________________2. { 2, 4, 6, 8, 10}
________________3{ 2, -4, 6, -8, 10, …}
________________4. { 3, 6, 9, 12, 15 }
________________5. { 1, 4, 7, 10, 13 }

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