Sequences and Series

Hello!
How are you? Are you prepared?

2
3
Study Outline

Definition of Sequences and Series

●Arithmetic ●Harmonic ●Geometric

○Sequence ○Sequence ○Sequence
○Mean ○Mean ○Mean
○Series ○Series ○Series

4
Study Outline

●Fibonacci Sequence

●Binomial Expamsion

5
 1
Definition of Sequence and
Series
Let’s start with the first set of slides
 Sequence
Def’n: succession of numbers formed by fixed rules denoted by:

𝑎1 , 𝑎2 , 𝑎3 , … , 𝑎 𝑛 ,…
 

Examples

Natural Numbers 1, 2, 3, 4, 5,….

Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13,…

7
krladaga 2020
Two Kinds

● Finite – the sequence ends at the nth term

  𝑁
{𝑎 }
𝑖 𝑖=1
● Infinite - the sequence goes on to infinity

  ∞
{𝑎 }
𝑖 𝑖=1

8
Series
Def’n: indicated sum of a sequence of numbers denoted by:

 
𝑆 𝑛 =𝑎1 +𝑎 2+…+ 𝑎𝑛

9
General Term of a Sequence/Series

An expression involving n such that by

substituting n = 1, 2, 3, … , one obtains the first,
second, third, and nth term of the sequence or
series

10
 2
Arithmetic Sequences,
Mean, and Series
How are you?
Arithmetic
Sequences
<basics>

12
Arithmetic Sequence
Def’n: sequence where the difference between terms is a constant
(common difference)

4 , 7, 10, 13, 16, …

+3 +3 +3 +3
(common difference)

13
General Term of Arithmetic Sequence

𝑎 𝑛=𝑎 1+ 𝑑 (𝑛 − 1)
 

where an = nth term

a1 = first term
d = common difference

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Sample Problems

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16
17
18
19
20
21
N-th difference

Original Sequence

First difference

Second difference

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Solving Technique (up to 3rd Difference)

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 Solving Technique (up to 3rd Difference)

General Term of a Quadratic/Cubic Sequence

3 2
𝑎 𝑛= 𝑤 𝑛 + 𝑥 𝑛 + 𝑦𝑛+ 𝑧
 

24
Solving Technique (up to 3rd Difference)

 
𝑤 + 𝑥+ 𝑦 + 𝑧= 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
7 𝑎+3 𝑏+ 𝑐=𝑓𝑖𝑟𝑠𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
 

12 𝑎+2 𝑏=𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒

 

6 𝑎=𝑡h𝑖𝑟𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒
 

25
26
{
 
𝑤 + 𝑥+ 𝑦 + 𝑧= 4
 
7 𝑤 +3 𝑥 + 𝑦=10
 
12 𝑤+2 𝑥=16
6 𝑤=6
 

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 EQUATION 1:
6  𝑤=6 w=1
  /3

EQUATION 2: EQUATION 3: EQUATION 4:

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  𝑤+2 𝑥=0 7  𝑤 +3 𝑥 + 𝑦=2 𝑤
  + 𝑥+ 𝑦 + 𝑧=0
12(1/
  3)+2 𝑥=0 7
  ( 1/ 3 ) +3 ( − 2 ) + 𝑦=2 1/
  3+(− 2)+(17 / 3)+ 𝑧 =0
2  𝑥 =0 − 4 7/
  3+(− 6)+ 𝑦=2 4
  + 𝑧=0
2  𝑥 =− 4  
𝑦=2 − +6
7
𝑧=−
  4
𝑥=−2
  3
 𝑦=17 / 3

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 3 2
∴ 𝑎 𝑛=𝑛 +2 𝑛 − 3 𝑛+ 4
 

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Arithmetic Mean
<basics>

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Arithmetic Mean
Def’n: set of values is the ratio of their sum to the total number of
values in the set.

 
𝑥 1+ 𝑥 2+ …+ 𝑥 𝑛
𝑚=
𝑛

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Arithmetic Series
<bit more difficult>

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Arithmetic Series
Def’n: indicated sum of the first few terms of an arithmetic
sequence

  𝑛
𝑆 𝑛 = (𝑎 1+ 𝑎𝑛 )
2
  𝑛
𝑆 𝑛 = [ 2 𝑎1 + ( 𝑛 − 1 ) 𝑑 ]
2

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Proof

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35
GIVEN: Required:
n=5
an = 2n -5 Sum Of The First Five
Terms

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37
GIVEN: Required:
n=8
a1 = 3 Sum Of The First Eight
d = -2 Terms

38
39
 3
Geometric Sequences, Mean,
and Series
Have you eaten?
Geometric
Sequences
<basics>

41
Geometric Sequence
Def’n: sequence where the sequence in which each term after the
first is a constant multiple of the preceding term (constant ratio)

  𝑎 𝑛+1
𝑟=
𝑎𝑛

42
Geometric Sequence
Def’n: sequence where the sequence in which each term after the
first is a constant multiple of the preceding term (constant ratio)

2, 6, 18, 54, 162, …

  𝑎 𝑛+1 6 18 54 162
𝑟= = = = = =3
𝑎𝑛 2 6 18 54
43
General Term of Geometric Sequence

𝑛− 1
𝑎 𝑛=𝑎 1 𝑟
 

where an = nth term

a1 = first term
r = common ratio

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45
46
47
Geometric Mean
<basics>

48
Geometric Mean
Def’n: average value or mean which signifies the central
tendency of the set of numbers by finding the product of their
values.

𝑛
𝐺 . 𝑀 .=√ 𝑥1 𝑥2 … 𝑥 𝑛
 

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Geometric Series
<basics>

50
Geometric Series
Def’n: indicated sum of the first few terms in a geometric
sequences; signifies the central tendency of the set of numbers

 
𝑎1 (𝑟 𝑛 − 1)
𝑆𝑛= ( 𝑔𝑖𝑣𝑒𝑛 𝑟 ≠ 1)
𝑟 −1
 
𝑎1 (1− 𝑟 𝑛 )
𝑆𝑛=
1− 𝑟

51
Proof

52
Sample Problems

GIVEN: Required:
n=6
a1 = -2 Sum Of The First Six
r=3 Terms

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Sample Problems

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 4
Harmonic Sequences, Mean,
and Series
Let’s start with the first set of slides
Harmonic Mean
<basics>

56
Harmonic Mean
Def’n: reciprocal of the average of the reciprocals

  𝑛
𝐻 . 𝑀 .=
1 1 1
+ +…+
𝑥1 𝑥 2 𝑥𝑛

57
Harmonic Mean
Def’n: reciprocal of the average of the reciprocals

  𝑛
𝐻 . 𝑀 .=
1 1 1
+ +…+
𝑥1 𝑥 2 𝑥𝑛

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Harmonic
Sequences
<basics>

59
Harmonic Sequence
Def’n: a sequence of numbers such that their reciprocals form an
arithmetic sequence

 
1 1 1
1 ,2 , 3 , 4 , …
 
1, , , ,…
2 3 4

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Harmonic Sequence
Def’n: a sequence of real numbers such that any term in the
sequence is the harmonic mean of its two neighbors.

 
1 1 1
1, , , ,…
2 3 4

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Def’n: a sequence of real numbers such that any term in the
sequence is the harmonic mean of its two neighbors.

  2 1
=
1 1 3

[( ) ] [( )]
1
2
+
1
4

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Proof

63
Sample Problem

Given: a3 (harmonic) = 3 Equation:

a10 (harmonic) = 10  𝑎 =𝑎 1+ 𝑑 (𝑛 − 1)
𝑛

Required: n

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Solution: 𝑎
  𝑛=𝑎 1+ 𝑑 (𝑛 − 1)

1 1
𝑎 3 ( 𝑎𝑟𝑖𝑡h𝑚𝑒𝑡𝑖𝑐 )= 𝑎 10 ( 𝑎𝑟𝑖𝑡h𝑚𝑒𝑡𝑖𝑐 )=
3 10

𝑎
  3 =𝑎 1+ 𝑑 (3− 1) 𝑎
  10 =𝑎1 +𝑑 (10 − 1)

1
  =𝑎 +2 𝑑  1 =𝑎 +9 𝑑
1 1
3 10

65
Solution:

1
  1 +9 𝑑 =
𝑎
10
-   1 +2 𝑑 = 1
𝑎  𝑎1 + 2 (−
30
1
) =
1
3
3
−2 1
+(
30 )
 𝑎1 =
7 3
  𝑑=−
7
30

1 2
  =−
𝑑   1=
𝑎
30 5

66
 2 1
Solution: General Term 𝑎  𝑛= − ( 𝑛− 1 )
5 30

n where an = 0 @ n=13:
a13 (arithmetic)= 0
  2 1
0= − ( 𝑛 − 1) a13 (harmonic)= 1/0
5 30

  2 − 1 𝑛+ 1
0=
5 30 30
n = 12 FINAL ANSWER
  2 − 1 𝑛+ 1
0=
5 30 30

 1 𝑛= 2 + 1
30 5 30 n = 13
NOT FINAL ANSWER

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Harmonic Series
<basics>

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General Harmonic Series
Def’n:  
∞
1
∑ 𝑎𝑛+𝑏
𝑛=1
 

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Harmonic Series
Def’n:
 
∞
1 1 1
∑ 𝑛 =1+ 2 + 3 + …
𝑛=1

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 Maximum Partial Sum of Harmonic Series

Def’n:
 
𝑘

∑ 𝑎𝑛
𝑛=1
  𝑤h𝑒𝑟𝑒 𝑎 𝑛 ≥ 0

71
Sample Problem

 1 1 1 1
, , , … , 1, −1 , −
19 17 15 3

Maximum Partial Sum

72
Sample Problem

 1 1 1
+ + + …+1 ≅ 2.13
19 17 15

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 Sum of Terms of Harmonic Sequence
(approximation)
Def’n:

 
1 2 𝑎 1+ ( 2 𝑛 − 1 ) 𝑑
𝑆 𝑛 ≈ 𝑙𝑛
𝑑 (
2 𝑎1 − 𝑑 )
𝑤h𝑒𝑟𝑒
  2 𝑎 ≠ 𝑑 𝑎𝑛𝑑 𝑑 ≠ 0

74
 

75
 
Sample Problem

Given: a1 (arithmetic) = 3 Equation:

d=2 2 𝑎 1+ ( 2 𝑛 − 1 ) 𝑑
  1
n=9
Required: Sn
𝑆𝑛 ≈
𝑑
𝑙𝑛
( 2 𝑎1 − 𝑑 )

76
Solution:
  1 2 𝑎1 + ( 2 𝑛 − 1 ) 𝑑
𝑆 𝑛 ≈ 1+
( 𝑑
𝑙𝑛
2𝑎 1 − 𝑑 )
  1 2(3)+ ( 2 ( 9 ) − 1 ) (2)
𝑆 𝑛 ≈ 1+ ( 2
𝑙𝑛
2 (3) − (2) )
  1 ( 6+ 17 ) (2)
𝑆 𝑛 ≈ 1+
( 2
𝑙𝑛
6− (2) )
  1 40
𝑆 ≈ 1+ 𝑙𝑛 (
𝑛
2 4 )
Reasonable approximation to exact value
𝑆
  𝑛 ≈ 2.15 (≈2.13)
 4.5
Applications of Different
Sequences and Series
Let’s start with the first set of slides
Summations
<basics>

79
Arithmetic Series
Def’n:
 
𝑛

∑ (𝑘𝑖+𝑐)
𝑛=1
𝑤h𝑒𝑟𝑒 𝑘 𝑎𝑛𝑑 𝑐 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
 

80
Sample Problem

81
Geometric Series
Def’n:
 
𝑛
𝑖
∑ 𝑎 .𝑏
𝑛=1
𝑤h𝑒𝑟𝑒 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
 

82
Sample Problem

83
Business
Applications
<basics>

84
Arithmetic Sequence

Given: a1 = 3000 Equation:

d = 500
n = 24
𝑎 𝑛=3000+500 𝑛
 

85
Solution:
𝑎
  𝑛=3000+500 𝑛

 𝑎24 =3000+500(24)
 𝑎 =3000+12000
24

𝑎24 =15000
   

86
Annuity
(Geometric Sequences)
<basics>

87
Definitions
Annuity - a sequence of equal payments made at equal periods of time

ordinary annuity - payments are made at the end of the time period, and the
frequency of payments is the same as the frequency of
compounding,

payment period - the time between payments

Term of the annuity - time from the beginning of the first payment period to the
end of the last period

88
Definitions
future value of the annuity – defined as the sum of the compound amounts of all
the payments, compounded to the end of the term.

  𝑛
(1+𝑖) −1
𝑆= 𝑅
𝑖 [ ]
where S = future value
R = payment at the end of each period
i = interest rate per period
n = number of periods
89
Sample Problem

Given: R = 22 000 Equation:

I = 0.06
  𝑛
(1+𝑖) − 1
n=7
𝑆= 𝑅 [
𝑖 ]
90
 𝑛
  (1+𝑖) − 1
Solution: 𝑆= 𝑅[ 𝑖 ]
7
  (1+0.06)   −1
𝑆=22000 [ 0.06 ]
𝑆=184,664.
  43

91
Sample Problem

92
Practice Problems
<basics>

93
Determine if sequence is arithmetic. Find indicated term using given
If YES, name common difference. information.
If NO, determine the pattern that
forms the sequence 1.
  𝑎1 , =5 , 𝑑= 4 , 𝑓𝑖𝑛𝑑 𝑎 15
2.
  𝑎1 , =9 , 𝑑 =−2 𝑓𝑖𝑛𝑑 𝑎 17
1. -5, -2, 1, 4, 7, 10….
2. 1, 4, 8, 13, 19 ,26, … Find number of terms in the
3. -5, -2, 1, 4, 7 … sequence using given information.

Find the general form for the 1.

  𝑎1 , =2 , 𝑎𝑛 =−22 , 𝑑 =−3 ,
arithmetic sequence and find the 1.
  𝑎1 , =4 , 𝑎𝑛 =−, 𝑑 =−3 ,
100th term

1. 6, 15, 24, 33, …

2. 16, 14.5, 13, …
3. 7, 4, 1, -2, -5,. ..

94
 4

Fibonacci Sequence
Let’s start with the first set of slides
 4

Fibonacci Sequence
Let’s start with the first set of slides
 6

Binomial Expansion
Let’s start with the first set of slides
98
You can also split
your content
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Is the color of milk and fresh snow, the Is the color of ebony and of outer space.
color produced by the combination of all It has been the symbolic color of
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99
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columns
Yellow Blue Red
Is the color of gold, butter Is the colour of the clear sky Is the color of blood, and
and ripe lemons. In the and the deep sea. It is located because of this it has
spectrum of visible light, between violet and green on historically been associated
yellow is found between the optical spectrum. with sacrifice, danger and
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Whoa! That’s a big number, aren’t you proud?

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That’s a lot of money

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108
Let’s review some concepts
Yellow Blue Red
Is the color of gold, butter and ripe Is the colour of the clear sky and Is the color of blood, and because
lemons. In the spectrum of visible the deep sea. It is located between of this it has historically been
light, yellow is found between violet and green on the optical associated with sacrifice, danger
green and orange. spectrum. and courage.

Yellow Blue Red

Is the color of gold, butter and ripe Is the colour of the clear sky and Is the color of blood, and because
lemons. In the spectrum of visible the deep sea. It is located between of this it has historically been
light, yellow is found between violet and green on the optical associated with sacrifice, danger
green and orange. spectrum. and courage.

109
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