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Principles of Mathematics 12-Chapter 2
Terry Fox Math 2007 1
CHAPTER 2 SEQUENCES AND SERIES OUTLINE
Day Section Topic
1 2.7 Geometric Sequences
2 2.8 Geometric Series
3 2.9 Infinite Geometric Series and Sigma
Notation
4 Review
5 Review
6 Chapter 2
Test
Principles of Mathematics 12-Chapter 2
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2.7 GEOMETRIC SEQUENCES
Today we will review geometric sequence from grade 10 math.
We will now be able to solve for the number terms in a sequence
by using logarithms.
The parts of a geometric sequence:
Example: 3, 6, 12, 24, 48, 96, 192, 384
The first term, 3 is called a or 1T . The second term, 6 is called 2T .
The number of terms in the sequences, 8 is called n.
Any term divided by its preceding term is called the common
ratio, r. In this example, r is 2.
The equation representing the value of any term is: 1 n
n arT .
Working with 1 n
n arT :
Given the sequence 2, 6, 18, 54,……., determine the value of 9T .
Principles of Mathematics 12-Chapter 2
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Given the sequence 100 000, 80 000, 64 000,…………32 768,
what is the term number of 32 768?
Given 21 T and 3936610 T , determine the value of the
common ratio.
In a geometric sequence, determine the single geometric mean
between 4 and 196.
Principles of Mathematics 12-Chapter 2
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A ball is dropped from a height of 8 meters. It rises to 70% of its
previous height after each bounce. What is the maximum height
that the ball will reach after it has bounced five times?
Assignment: 2.7 page 115
#1-7, 9, 10
Principles of Mathematics 12-Chapter 2
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2.8 GEOMETRIC SERIES
Last day we worked with geometric sequences. Today we will work
with geometric series. A series is the sum of geometric sequence.
Example: 3, 6, 12, 24, 48, 96, 192 this is a geometric sequence
3+6+12+24+48+96+192 this is a geometric series
Terminology for the series above:
311 STa 62 T 9632 S 2r
Determining the Sum of a Geometric Series:
Given the series: 2+6+18+54+162
We will call S the sum
of the series, therefore:
If we multiply this by 3
(The common ratio) we get:
Principles of Mathematics 12-Chapter 2
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This works for any geometric series. From this knowledge we can
develop a formula.
1432 ....... n
n arararararaS
nrS nn arararararar 1432 .......
r
raS
n
n
1
1 is the formula for the sum of any geometric series.
This formula is given on the provincial exam formula sheet.
Another useful formula is r
rlaSn
1 where l represents the last
term. This is also on the provincial exam formula sheet.
Principles of Mathematics 12-Chapter 2
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Applying the formulae:
1. Determine the sum of the first 8 terms of the series: 3282 …….
2. Determine the sum of the series: 8
1...........163264
Principles of Mathematics 12-Chapter 2
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3. The first term of a geometric series is 3 and the sum of the series
is 1533. How many terms are in the series if the common ratio
is 2?
4. The sum of a geometric series is 7812 and the common ratio
is 5 . What is 1T if there are six terms in the series?
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5. For a given series, 2
33 1
n
nS , determine 3T .
6. A patient is prescribed a medicine for an infection. She must
take 125mg on the first day and take half of the previous days
dosage for nine days. How much medicine has she taken by
the end of her treatment?
Principles of Mathematics 12-Chapter 2
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7. A ball is dropped from a height of 6 meters and bounces back
to a height of 75% of the previous height? What is the total
vertical distance that the ball has traveled after the 5th bounce?
Assignment: 2.8 page 124
#1-10, 13, 14
Principles of Mathematics 12-Chapter 2
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2.9 INFINITE GEOMETRIC SERIES
Today we will learn how to find the sum of infinite geometric series
and learn how to work with series written in sigma notation. Sigma
notation is a shorthand expression for a series.
Finding the sum of an infinite series:
Notice the pattern of the following series
4
1
2
113S
8
1
4
1
2
114S .
16
1
8
1
4
1
2
115S
Notice that with a common ratio of 2
1, the more that we multiply by
the common ratio by itself, the closer it gets to be equal to zero.
From the formula
r
raS
n
n
1
1
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Therefore:
For any infinite series with a common ratio 10 r we can have a
finite sum and the formula
that we can use is: r
aS
1
If the absolute value of r does not fit the restriction, then we cannot
determine a sum.
Applications:
1. Determine the sum of the series: 2
33612 …………
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2. An infinite geometric series has a finite sum. If the common
ratio is 1x , what are the possible values for x ?
3. The first term of an infinite geometric series is 10 and the sum to
infinity is 30. What is the common ratio?
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4. The sum to infinity of a series is 5
12 and the common ratio is
4
3.
What is the first term?
Sigma Notation:
Sigma notation is shorthand for “the sum of”.
Given:
5
2
123
n
n we can expand this series by subbing the values
of n starting at 2 and continuing until it becomes 5.
1514131223232323
which becomes 4824126
Notice that there were 4 terms. We can always determine the
number of terms in the series by subtracting the bottom number
from the top.
4125
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Also notice that the common ratio is 2. It can be determined by
looking at the power or by dividing
term 2 by term 1. If you are ever in doubt about the common ratio,
it can be determined this way.
Determining the sum:
110
4
35
n
n
12
113
2
kk
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Try these:
1. Determine the number of terms in 118
5
43
n
n
2. Determine the first term and the common ratio in
5
1
2
4
3
k
k
3. Determine the sum for
2
1
3
112
n
n
Principles of Mathematics 12-Chapter 2
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4. Determine the sum for
n
k
k
1
2
5. Determine the sum for
4
1
2logn
n
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6. A ball is dropped from a height of 6 meters and bounces back
to a height of 75% of the previous height? What is the total
vertical distance that the ball has traveled after the ball has
come to rest?
Assignment: 2.9 page 130 and Sigma Worksheet
#1-6 (2.9) and all from the worksheet