Chapter 11 Sec 3

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Chapter 11 Sec 3

• Geometric Sequences

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Geometric Sequence

• A geometric sequence is a sequence in which each
  term after the first is found by multiplying the
  previous term by a constant r called the common
  ratio.

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Example 1
Find the eighth term of a geometric sequence for
which a1 3 and r 2. an a1 r n 1
a8 (3) (2) 8 1 a8 (3)
(128) a8 384
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Example 2
Write an equation for the n term of a geometric
sequence 3, 12, 48, 192 an a1 rn 1
an (3) (4) n 1 So the equations is an
3(4) n 1
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Example 3
Find the tenth term of a geometric sequence for
which a4 108 and r 3. an a1 r n 1
a4 a1 (3) 4 1 108 a1 (3) 3
108 27a1 4 a1
an a1 r n 1 a10 4 (3) 10 1
a10 4 (3) 9 a10 78,732
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Geometric Means
As we saw with arithmetic means, you are given
two terms of a geometric sequence and are asked
to find the terms between, these terms between
are called geometric means. Find the three
geometric means between 3.12 and 49.92. 3.12,
_____, _____, _____, 49.92 an a1 r n
1 a5 3.12 r 5 1 49.92 3.12 r 4 16 r
4 2 r So
24.96 24.96
6.24 6.24
12.48
a1 a2 a3 a4
a5
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Chapter 11 Sec 4

• Geometric Series

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Geometric Series
Geometric Sequence Geometric Series. 1, 2, 4,
8, 16 1 2 4 8 16 4, 12, 36 4 (12)
36 Sn represents the sum of the first n terms
of a series. For example, S4 is the sum of the
first four terms.
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Example 1
Evaluate
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Only have the first and last terms?
You can use the formula for finding the nth term
in (an a1 r n 1 ) conjunction with the sum
formula when you dont
know n. an r a1 r n 1 r an r a1
r n
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Example 3
Find a1 in a geometric series for which S8
39,360 and r 3.
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Chapter 11 Sec 5

• Infinite Geometric Series

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Infinite Geometric Series
Any geometric series with an infinite number of
terms. Consider the infinite geometric series
You have already learned to find the sum Sn of
the first n terms, this is called partial sum for
an infinite series.
Notice that as n increases, the partial sum
levels off and approaches a limit of one. This
leveling-off behavior is characteristic of
infinite geometric series for which r lt 1.
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Sum of an Infinite Series
Lets use the formula for the sum of a finite
series to find a formula for an infinite
series. If 1 lt r lt 1 , the value if rn will
approach 0 as n increases. Therefore the partial
sum of the infinite series will approach
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Example 1
Find the sum of each infinite geometric series,
if it exists.
First find the value of r to determine if the
sum exists.
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Example 2 Sigma Time
Evaluate
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Repeeeeating Decimal
typo intentional

• Write 0.39 as a fraction.
• S 0.39
• S 0 .393939393939 then
• 100S 39.393939393939 Subtract 100S S
• S 0 .393939393939
• 99S 39

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Daily Assignment

• Chapter 11 Sections 3 5
• Study Guide (SG)
• Pg 145 150 Odd