1. SIGMA NOTATIOM, SEQUANCES AND SERIES

2. MAIN TOPIC
4.1 Sequences and Series
4.2 Sigma Notation
4.3 Arithmetic Sequences and Series
4.3 Geometric Sequences and Series
4.4 Infinite Geometric Series

3. OBJECTIVE
At the end of this topic you should be able to
Define sequences and series
Understand finite and infinite sequence, finite and infinite series
Use the sum notation to write a series

4. SEQUENCES and SERIES
A sequence is a set of real numbers a1, a2,…an,… which is arranged (ordered).
Example:
Each number ak is a term of the sequence.
We called a1 - First term and a45 - Forty-fifth term
The nth term an is called the general term of the sequence.

5. INFINITE SEQUENCES INFINITE SEQUENCES
An infinite sequence is often defined by stating a formula for the nth term, an by using {an}.
Example:
The sequence has nth term
Using the sequence notation, we write this sequence as follows
INFINITE SEQUENCES
First three terms
Fifth teen term

6. EXERCISE 1 : Finding terms of a sequence
List the first four terms and tenth term of each sequence: A B C D E F

7. Definition of Sigma Notation
Consider the following addition :
Based on the patterns of the addends, the addition above can be written in the following form
=(3(1) – 1) + (3(2) – 1) + (3(3) – 1) + (3(4) – 1) + (3(5) – 1) + (3(6) – 1)

8. The amount of the term in the addition above can be written as (3i – 1). The term in the addition are obtained by substituting the value of i with the value of 1, 2, 3, 4, and 5 to (3i – 1)
The symbol read as sigma, is used to simplify the expression of the addition of number with certain patterns. In order that you understand more, the addition above can be written as : =
=(3(1) – 1) + (3(2) – 1) + (3(3) – 1) + (3(4) – 1) + (3(5) – 1) + (3(6)-1)
=(3(1) – 1) + (3(2) – 1) + … + (3(i) – 1) + … + (3(6)-1)

9. In general, the sum of n term of number with certain pattern where the ith term is stated as Ui can be written as :
U1 + U2 + U3 + … + Ui + …+ Un =
Where :
i = 1 is the lower bound of the addition
n is the upper bound of the addition

10. Example 1 :
Change the following of addition in the form of sigma notation! a b

11. Answer :
= 3(1) + 3(2) + 3(3) + 3(4) + 3(5) + 3(6) + 3(7)
= 3(1) + 3(2) + … + 3(i) + … + 3(7) =
= (4(1) – 3) + (4(2) – 3) + (4(3) – 3) + … + (4(6) – 3)
= (4(i) – 3)

12. Determine the value of the following addition that state in sigma notation! a. b.

13. Answer :
= (2(1) – 3) + (2(2) – 3) + (2(3) – 3) + (2(4)-3) = (2 – 3) + (4 – 3) + (6 – 3) + (8 – 3) = = 8 = = (1 + 1)+(4 + 1)+(9 + 1)+(16 + 1)+(25 + 1)+(36 +1) = = 66

14. THEOREM OF SUMS
Sum of a constant
Sum of 2 infinite sequences

15. Arithmetic Sequence and Series

16. OBJECTIVE Recognize arithmetic sequences and series
At the end of this topic you should be able to :
Recognize arithmetic sequences and series
Determine the nth term of an arithmetic sequences and series
Recognize and prove arithmetic mean of an arithmetic sequence of three consecutive terms a, b and c

17. THE nth TERM OF AN ARITHMETIC SEQUENCES
An arithmetic sequence with first term a and common different b, can be written as follows:
The nth term, an of this sequence is given by the following formula:
a, a + b, a + 2b, … , a + (n – 1)b
Un = a + (n – 1)b

18. ARITHMETIC SEQUENCES b = U2 – U1
A sequence U1, U2,…Un,… is an arithmetic sequence if there is a real number b such that for every positive integer k,
The number is called the common difference of the sequence.
U2 – U1 = U3 – U2 = … = Un – Un-1 = a constant
b = U2 – U1

19. Find the formula for the and term if given the arithmetic sequence :
Example 1:
Find the formula for the and term if given the arithmetic sequence :
A , 4, 7, 10, …
B , 48, 43, …

20. Answer : A 1, 4, 7, 10 Based on the sequence, then obtained : a = 1,
b = U2 – U1
= 4 – 1 = 3
Un = a + (n – 1)b
= 1 + (n – 1)3
= 1 + 3n – 1
= 3n
U30 = 3n
= 3(30)
= 90

21. Answer : B 53, 48, 43, … Based on the sequence, then obtained :
b = U2 – U1
= 48 – 53 = -5
Un = a + (n – 1)b
= 48 + (n – 1)-5
= 48 + (-5n + 5)
= 48 – 5n + 8
= 53 – 5n
U30 = 53 – 5n
= 53 – 5(30)
= 53 – 150 = -97

22. EXERCISE 9: Finding a specific term of an arithmetic sequence
If given the arithmetic sequence U6 = 50 and U41 = 155 determine the twelfth.
If the fourth term of an arithmetic sequence is 5 and the ninth term is 20, find the sixth term.

23. Answer :
A U6 = 50, U21 = 155, U12?
U6 = a + 5b = 50
U21 = a + 20b = 155
-15b = -105
b = 7
a + 5b = 50
a + 5(7) = 50
a =
a = 15
U = a + (n – 1)b
= 15 + (29)7 =
= 218

24. Answer : A U4 = 5, U9 = 20, U6? U4 = a + 3b = 5 U9 = a + 8b = 20
U = a + (n – 1)b
= -4 + (5)3 =
= 11

25. Formula for the middle Term of on Arithmetic Sequence

26. Example :
Given an arithmetic sequence of 3, 8, 13, … , 283. Determine the middle term of the sequence. Which term is the middle term

27. Answer :
From the question it is know that U1 = 3 and Un = 283 So the middle term is Ut = 143 Ut = 143 a + (t – 1)b = (t – 1)5 = 143 (t – 1)5 = 140 t – 1 = 28 t = 29

28. THE nth PARTIAL SUM OF AN ARITHMETIC SEQUENCES
If a1, a2,…an,… is an arithmetic sequence with common difference b, then the nth partial sum Sn (that is the sum of the first nth terms) is given by either or

29. OBJECTIVE At the end of this topic you should be able to
Recognize geometric sequences and series
Determine the nth term of a geometric sequences and series
Recognize and prove geometric mean of an geometric sequence of three consecutive terms a, b and c
Derive and apply the summation formula for infinite geometric series
Determine the simplest fractional form of a repeated decimal number written as infinite geometric series

30. GEOMETRIC SEQUENCES
A sequence U1, U2,…Un,… is a geometric sequence if U1 ≠ 0 .
The number is called the common ratio of
the sequence.

31. Example :
Given Geometric sequence of 24, 12, 6, 3, …, where the formula of the nth term is Un . Determine Un and the sixth term of the sequence

32. Answer:
The geometric sequence of 24, 12, 6, 3, … , the first term of the sequence a = 24 and the ratio r= ½ . The formula for the nth term is :

33. THE SUM OF AN INFINITE GEOMETRIC SERIES
If |r| < 1 , then the infinite geometric series
has the sum

34. THE nth PARTIAL SUM OF AN GEOMETRIC SEQUENCES
If a1, a2,…an,… is a geometric sequence with common ratio r ≠ 0 , then the nth partial sum Sn (that is the sum of the first nth terms) is given by
if r < 1
if r > 1

35. Example :
Given that a geometric sequence : 2, 6, 18, 54, …, Un, Determine :
a. The formula for nth term and
b. The sum of the sixth n term!

36. Answer Sequence a geometric sequence : 2, 6, 18, 54, … Un
Have a = 2 and r = 3
The formula for nth term is
The Sum of the first n term is :

37. THE SUM OF AN INFINITE GEOMETRIC SERIES
If |r| < 1 , then the infinite geometric series
has the sum

38. EXERCISE 17: Find the sum of infinite geometric series
The following sequence is infinite geometric series. Find the sum

39. Answer :
i) a = 2 ,

40. APPLICATIONS OF ARITHMETIC AND GEOMETRIC SERIES

41. OBJECTIVE
At the end of this topic you should be able to
Solve problem involving arithmetic series
Solve problem involving geometric series

42. APPLICATION 1: ARITHMETIC SEQUENCE
A carpenter whishes to construct a ladder with nine rungs whose length decrease uniformly from 24 inches at the base to 18 inches at the top. Determine the lengths of the seven intermediate rungs.
a1 = 18 inches
a9 = 24 inches
Figure 1

43. APPLICATION 2: ARITHMETIC SEQUENCE
The first ten rows of seating in a certain section of stadium have 30 seats, 32 seats, 34 seats, and so on. The eleventh through the twentieth rows contain 50 seats. Find the total number of seats in the section.
Figure 2

44. APPLICATION 1: GEOMETRIC SEQUENCE
A rubber ball drop from a height of 10 meters. Suppose it rebounds one-half the distance after each fall, as illustrated by the arrow in Figure 3. Find the total distance the ball travels. 10
Figure 3

45. APPLICATION 2: GEOMETRIC SEQUENCE
If deposits of RM100 is made on the first day of each month into an account that pays 6% interest per year compounded monthly, determine the amount in the account after 18 years.
Figure 4