Transcript A. Our Lives are Sequences and Series

A. Our Lives are
Sequences and Series
Pre-Calculus 20
P20.10
Demonstrate understanding of
arithmetic and geometric (finite and
infinite) sequences and series.
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Key Terms:
Fibonacci Sequence
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The Fibonacci Sequence is often called
Nature’s Numbers because it occurs so
often in nature.
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1,1,2,3,5,8,13,21,…….
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What is the next term in the pattern?
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This spiral pattern formed by the FS is
found in the inner ear, star clusters,
clouds, whirl pools, pedals of flowers,
etc.
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We will be looking at two different types
of sequences in this unit.
1. Arithmetic Sequences
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P20.10
Demonstrate understanding of arithmetic
and geometric (finite and infinite)
sequences and series.
1. Arithmetic Sequences
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A sequence is an ordered list of objects.
It contains elements or terms that follow
a pattern or rule to determine the next
term
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Each term in the sequence is labeled
according to its position in the sequence.
= 1st term
n = number of terms
tn = a general tern in the sequence
 t1
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Finite sequences have a finite number of
terms: 2,5,8,11,14
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Infinite sequences have a infinite number
of terms: 5, 10, 15, …….
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An Arithmetic Sequence is an ordered
list of terms in which the difference
between consecutive terms is constant.
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So the same value or variable is added
or subtracted each time to create the
next term. This is called the Common
Difference.
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To get the Common Difference you
subtract any term by the term directly in
front of it.
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The General Term Formula allows us to
determine the value of any term in any
AS.
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Consider the AS: 10, 16, 22, 28
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We can rewrite the formula as:
Example 1
Example 2
Example 3
Example 4
Key Ideas
p. 16
Practice
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Ex. 1.1 (p.16) #1-3, 6-17
#8-24 evens, 25,26
2. Adding Up a Sequence (1)
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P20.10
Demonstrate understanding of arithmetic
and geometric (finite and infinite)
sequences and series.
2. Adding Up a Sequence (1)
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His method is referred to as an
Arithmetic Series which is a short way of
adding together all the terms in a
sequence
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The sum of an arithmetic series can be
determined using the following formula:
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We can also adapt the formula by
subbing tn in for the general term of the
sequence.
Example 1
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Determine the number of flashes in 1st
42 minutes.
Example 2
Key Ideas
p. 27
Practice
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Ex. 1.2 (p.27) #1-6 odds in each, 7-15
#7-20
3. Geometric Sequences
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P20.10
Demonstrate understanding of arithmetic
and geometric (finite and infinite)
sequences and series.
3. Geometric Sequences
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Investigate
p. 33
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In a Geometric Sequences the ratio of
consecutive terms is constant.
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The Common Ratio, r, can be found by
dividing any term by the term in front of it
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The General Term Formula for GS:
Example 1
Example 2
Example 3
Example 4
Key Ideas
p. 39
Practice
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Ex. 1.3 (p.39) #1-3, 6-17
#8-20 evens, 22-25
4. Adding Up a Sequence (2)
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P20.10
Demonstrate understanding of arithmetic
and geometric (finite and infinite)
sequences and series.
4. Adding Up a Sequence (2)
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A Geometric Series is the expression for
the sum of the terms of a Geometric
Sequence
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Find the sum of the 1st 5 terms of the
following GS
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3, 6, 12
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Easy Right?! What if I asked for the first
100 terms?
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We use the Geometric Series Formula:
Example 1
Example 2
Example 3
Key Ideas
p. 53
Practice
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Ex. 1.4 (p.53) #1-14
#9-22
5. Never Ending Geometric Series
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P20.10
Demonstrate understanding of arithmetic
and geometric (finite and infinite)
sequences and series.
5. Never Ending Geometric Series
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Investigate
p. 58
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Convergent Series
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As the number of terms increases the
sum of the series approaches a fixed
value of 8. Therefore the sum is 8.
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This is called a convergent series.
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Divergent Series
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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.
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This is called a divergent series.
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The Formula for the Infinite GS:
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Apply to 4+2+1+0.5+0.25+….
Example 1
Example 2
Key Ideas
p.63
Practice
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Ex. 1.5 (p.63) #1-16
#6-21