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Chapter 11
Sequence and Series
Lesson 11-1
Mathematical Patterns
Sequence

A sequence is an ordered list of numbers
called terms.
Example – Page 591, #2
Describe each pattern formed. Find the next three terms
4,8,16,32,64,... 128, 256,512
The pattern is to multiply by 2
Example – Page 591, #4
Describe each pattern formed. Find the next three terms
1, 4,7,10,13,...
16,19, 22
The pattern is to add 3
Recursive Formula

Recursive formula gives the first term and defines the
other terms in a sequence by relating each term to
the one before it.
 Example 1, 4, 7, 10, 13, …
1st term 2nd term
a1
a2
3rd term n – 1 term nth term
a3
an1
an
Example 3 – Page 591, #12
Write a recursive formula for each sequence. Then find
the next term.
The pattern is to add 1
2, 1,0,1, 2,...
a1  2
a2  a1  1  2  1  1
an  an1  1
a3  a2  1  1  1  0
a6  a5  1  2  1  3
a4  a3  1  0  1  1
a5  a4  1  1  1  2
Example 3 – Page 591, #16
Write a recursive formula for each sequence. Then find
the next term.
9
144,36,9, ,....
4
a1  144
a1 144
a2  
 36
4
4
a2 36
a3 

9
4
4
The pattern is to divide by 4
an1
an 
4
9
a4 4 9
a5 
 
4 4 16
Explicit Formula

Explicit formula expresses the nth term in a
sequence in terms of n, where n is a
positive integer.
Example – Page 591, #18
Write an explicit formula for each sequence. Then find a12
4,5,6,7,8,....
a1  4  1  3
a2  5  2  3
a3  6  3  3
The pattern is to add 1
an  n  3
a12  12  3  15
Example – Page 591, #20
Write an explicit formula for each sequence. Then find a12
4,7,10,13,16,....
a1  4  3(1)  1
a2  7  3(2)  1
a3  11  3(3)  1
The pattern is to add 3
an  3n  1
a12  3(12)  1  37
Example – Page 591, #22
Write an explicit formula for each sequence. Then find a12
1
1
2 , 2, 1 , 1,....
2
2
The pattern is to add ½
1
5 1 6
a1  2   
2
2
2
26
a2  2 
2
1 3 3  6
a3  1 

2 2
2
n6
an 
2
12  6
a12 
3
2
Lesson 11-2
Arithmetic Sequence
Arithmetic Sequence
Arithmetic sequence the differences
between consecutive terms is constant.
 The difference is called the common
difference.


The common difference can be positive (the
terms of the sequence are increasing) or
negative (the terms of the sequence are
decreasing)
Example 1 – Page 596, #2
Is the given sequence arithmetic? If so, identify the common
difference.
10, 20,30, 40,...
Yes; 10
The pattern is to adding 10
Example 1 – Page 596, #4
Is the given sequence arithmetic? If so, identify the common
difference.
0,1,3,6,10,...
No
The pattern is to adding 1, 2, 3
Arithmetic Sequence
Formula

Recursive Formula
an  an1  d

Explicit Formula
an  a1  (n  1)d

n is the number of terms and d is the
common difference
Example 2 – Page 596, #12
Find the 32nd term of each sequence.
9, 8.7, 8.4,...
an  a1  (n  1)d
a32  9  (32  1)0.3
 0.3
a1  9
n  32
d  0.3
Arithmetic Mean
Arithmetic mean of any two numbers is the
average of the two numbers.
 For any three sequential terms in an
arithmetic sequence, the middle term is the
arithmetic mean of the first and third term.

Example 3 – Page 596, #22
Find the missing term of each arithmetic sequence.
...5, , 21,...
5  21
mean 
 13
2
Fibonacci Sequence
Fibonacci Sequence
You can find each term of the sequence
using addition, but the sequence is not
arithmetic.
 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …..


Where each term after the second one is
the sum of the two preceding terms.
Fibonacci in Nature
Fibonacci spirals occur frequently in nature
because the math used for can be related to
a Fibonacci sequence.
 Examples

Nautilus shells
 Snails
 Arrangement of seeds on flower plants.

Pine Cones
Here is a picture of an ordinary pinecone seen
from its base where the stalk connects to the tree.
Can you see the two sets of spirals? How many
are their in each set.
Here is a picture of an ordinary pinecone seen
from its base where the stalk connects to the tree.
Can you see the two sets of spirals? How many
are their in each set.



1 1
1
Golden Mean
2 2
1
3  1.5
The ratio of each term to the previous
2
one gradually converges to a limit of
5  1.667
3
1.618 which is called the Golden
8  1.6
Mean.
5
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …
13  1.625
8
The Golden Ratio 1.618 is called Phi
21  1.615
in tribute to the great Phidias, who
13
used the proportion frequently in his
34  1.619
sculpture.
21
55  1.618
34
Golden Rectangles
1,1,2,3,5,8,13,21,34,55,...
Golden Rectangle
Draw a Fibonacci Spiral
X- Ray picture of Seashells
Golden Ratio



Pythagoras (560 – 480
BC), the Greek geometer,
was especially interested in
the golden section, and
proved that it was the basis
for the proportion of the
human figure.
He showed that the human
body is built with each part
a definite golden proportion
of all the parts.
Pythagoras’ discoveries of
the proportions of the
human figure had a
tremendous effect on
Greek
Pythagoras detail, from School
Athens by Raphael
The classic subdivisions of the rectangle align
perfectly with major architectural features of the
structure. The Parthenon, was perhaps the best
example of a mathematical approach to art.
Parthenon – Acropolis, Athens
Golden Rectangles in Art
Lesson11-3
Geometric Sequences
Geometric Sequence
In a geometric sequence, the ratio between
consecutive terms is constant.
 This ratio is called the common ratio.

Example 1 – Page 602, #2
Is the given sequence geometric? If so, identify the common
ratio and find the next two terms.
1, 2,3, 4,...
No
The pattern is adding 1
Example 1 – Page 602, #10
Is the given sequence geometric? If so, identify the common
Ratio and find the next two terms.
2, 10,50, 250,....
Yes; 1250, – 6250
The pattern is multiplying – 5
Geometric Sequence
Formula

Recursive Formula

Explicit Formula
an  an1  r
an  a1  r

n1
n is the number of terms and r is the
common ratio
Example 2 – Page 603, #14
Write the explicit formula for each sequence. Then generate
the first five terms.
a1  0.0237, r  10
an  a1  r
n1
an  0.0237 10
n 1
0.0237, 0.237, 2.37, 23.7, 237
a1  0.237
r  10
Geometric Mean

You can find the geometric mean of any
two positive numbers by taking the positive
square root of the product of the two
numbers.
Example 3 – Page 603, #22
Find the missing term of a geometric sequence.
5, ,911.25,....
mean  5(911.25)  67.5
Lesson 11-4
Arithmetic Series
Series

A series is the expression for the sum of
the terms of a sequence.
Finite sequences (6, 9, 12, 15, 18) and
series (6 + 9 + 12 + 15 + 18) have terms
that you can count individually from 1 to a
final whole number.
 Infinite sequences (6, 9, 12, 15, 18, ….)
and series (6 + 9 +12 +15 +18 +…..)
continue without end.

Example 1 – Page 610, #2
Write the related series for each finite sequence. Then
evaluate each series.
5, 15, 25, 35, 45
5  (15)  (25)  (35)  (45)  125
Example 1 – Page 610, #6
Write the related series for each finite sequence. Then
evaluate each series.
4.5,5.6,6.7,.....,11.1
Adding 1.1
4.5  5.6  6.7  7.8  8.9  10  11.1  54.6
Sum of a Finite Arithmetic
Series

Arithmetic series is a series whose terms
form an arithmetic sequence. When you
know the first and last terms of the
sequence, you can use the formula to
evaluate the related series.
n
S n  (a1  an )
2
Example 2 – Page 610, #8
Each sequence has eight terms. Evaluate each related
series.
1, 1, 3,...., 13
n
S n  (a1  an )
2
8
s8  1  (13)   48
2
n8
a1  1
a8  13
Summation Notation


You can use summation symbol Σ (Greek letter sigma)
to write a series.
You can use limits to indicate how many terms you are
adding.
 Limits are the least and greatest integral values of n
Upper limit,
greatest value of n
Lower limit,
least value of n
3
  5n  1
n 1
Explicit formula
for the sequence
Example 3 – Page 610, #14
Use summation notation to write each arithmetic series for
the specified number of terms.
8  9  10  ....; n  8
a1  8  1  7
a2  9  2  7
a3  10  3  7
adding 1
8
n  7
n 1
Example 4 – Page 610, #20
For each sum, find the number of terms, the first term, and the
last term. Then evaluate the series.
5
  2n  1
n 1
n
S n  (a1  an )
2
5
S5   3  (11)   35
2
a1  2(1)  1  3
a5  2(5)  1  11
Lesson 11-5
Geometric Series
Sum of a Finite Geometric
Series

Geometric series is the expression for the
sum of the terms of geometric sequence.
Sn 
a1 1  r
1 r
n

Example 1 – Page 616, #4
Evaluate the series to the given term.
1
multiply – 6
 1  6  36  ....; S5
6
n
a1 1  r
Sn 
1 r
1
5
1  (6) 

1296.17
6
S5 

 185.17
1  (6)
7


1
a1 
6
r  6
n5
Infinite Geometric Series

In some cases you can evaluate an infinite
geometric series.
When | r | < 1, the series converges, or gets
closer and closer, to the sum S.
 When | r | ≥ 1, the series diverges, or
approaches no limit.

Sum of an Infinite Geometric
Series

An infinite geometric series with | r | < 1,
converges to the sum
a1
S
1 r
Example 3 – Page 616, #10
Decide whether each infinite geometric series diverges or
converges. State whether each series has a sum.
1 1
1    ....
2 4
1
multiply
2
r 1
1
1
2
The series converges and has a sum
Example 3 – Page 616, #12
Decide whether each infinite geometric series diverges or
converges. State whether each series has a sum.
1  2  4  ....
multiply 2
r 1
2 1
The series diverges and does not have a sum
Example 4 – Page 617, #18
Evaluate each infinite geometric series
1.1  0.11  0.011  ...
a1
S
1 r
1.1
S
 1.2
1  0.1
Dividing by a 10
r  0.10
a1  1.1