A. Our Lives are Sequences and Series

Pre-Calculus 20

P20.10

Demonstrate understanding of arithmetic and geometric (finite and infinite) sequences and series.

 Key Terms:

Fibonacci Sequence

 The Fibonacci Sequence is often called Nature ’ s Numbers because it occurs so often in nature.

 1,1,2,3,5,8,13,21,…….

 What is the next term in the pattern?

 This spiral pattern formed by the FS is found in the inner ear, star clusters, clouds, whirl pools, pedals of flowers, etc.

 We will be looking at two different types of sequences in this unit.

1. Arithmetic Sequences

 P20.10

 Demonstrate understanding of arithmetic and geometric (finite and infinite) sequences and series.

1. Arithmetic Sequences

 A sequence is an ordered list of objects. It contains elements or terms that follow a pattern or rule to determine the next term

 Each term in the sequence is labeled according to its position in the sequence.

  t 1 = 1 st term n = number of terms  t n = a general tern in the sequence

 Finite sequences have a finite number of terms: 2,5,8,11,14  Infinite sequences have a infinite number of terms: 5, 10, 15, …….

 An Arithmetic Sequence is an ordered list of terms in which the difference between consecutive terms is constant.

 So the same value or variable is added or subtracted each time to create the next term. This is called the Common Difference.

 To get the Common Difference you subtract any term by the term directly in front of it.

 The General Term Formula allows us to determine the value of any term in any AS.

 Consider the AS: 10, 16, 22, 28

 We can rewrite the formula as:

Example 1

Example 2

Example 3

Example 4

Key Ideas p. 16

Practice

 Ex. 1.1 (p.16) #1-3, 6-17 #8-24 evens, 25,26

2. Adding Up a Sequence (1)

 P20.10

 Demonstrate understanding of arithmetic and geometric (finite and infinite) sequences and series.

2. Adding Up a Sequence (1)

 His method is referred to as an Arithmetic Series which is a short way of adding together all the terms in a sequence

 The sum of an arithmetic series can be determined using the following formula:

 We can also adapt the formula by subbing t n sequence.

in for the general term of the

Example 1

 Determine the number of flashes in 1 st 42 minutes.

Example 2

Key Ideas p. 27

Practice

 Ex. 1.2 (p.27) #1-6 odds in each, 7-15 #7-20

3. Geometric Sequences

 P20.10

 Demonstrate understanding of arithmetic and geometric (finite and infinite) sequences and series.

3. Geometric Sequences

 Investigate p. 33

 In a Geometric Sequences the ratio of consecutive terms is constant.

 The Common Ratio, r, can be found by dividing any term by the term in front of it

 The General Term Formula for GS:

Example 1

Example 2

Example 3

Example 4

Key Ideas p. 39

Practice

 Ex. 1.3 (p.39) #1-3, 6-17 #8-20 evens, 22-25

4. Adding Up a Sequence (2)

 P20.10

 Demonstrate understanding of arithmetic and geometric (finite and infinite) sequences and series.

4. Adding Up a Sequence (2)

 A Geometric Series is the expression for the sum of the terms of a Geometric Sequence

 Find the sum of the 1 st following GS 5 terms of the  3, 6, 12

 Easy Right?! What if I asked for the first 100 terms?

 We use the Geometric Series Formula:

Example 1

Example 2

Example 3

Key Ideas p. 53

Practice

 Ex. 1.4 (p.53) #1-14 #9-22

5. Never Ending Geometric Series

 P20.10

 Demonstrate understanding of arithmetic and geometric (finite and infinite) sequences and series.

5. Never Ending Geometric Series

 Investigate p. 58

 Convergent Series

 As the number of terms increases the sum of the series approaches a fixed value of 8. Therefore the sum is 8.

 This is called a convergent series.

 Divergent Series

 As the sum of the terms increases, the sum of the series increases. The sum doesn ’ t approach a fixed value. Therefore the sum can not be calculated.

 This is called a divergent series.

 The Formula for the Infinite GS:

 Apply to 4+2+1+0.5+0.25+….

Example 1

Example 2

Key Ideas p.63

Practice

 Ex. 1.5 (p.63) #1-16 #6-21