CHAPTER 2 SEQUENCES AND SERIES OUTLINE Day Section Topicteachers.sd43.bc.ca/gbowers/Math12/Shared Documents/Chapter 2... · Principles of Mathematics 12-Chapter 2 ... CHAPTER 2 SEQUENCES

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



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 .



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.



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



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:



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.



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



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?



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?



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



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



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 …………



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?



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



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



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




n

k

k

1

2


4

1

2logn

n



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