Download - 2, 4, 8, 16, …
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2, 4, 8, 16, … 32Exercise
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2, 4, 6, 8, … Exercise
10
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1, 3, 9, 27, … 81Exercise
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1, , , , … 12
14
18
116
Exercise
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1, –2, 4, –8, 16, … –32Exercise
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3 6
x 2
12
x 2
24
x 2
48
x 2
96
x 2
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Geometric SequenceA geometric sequence is a sequence of numbers whose successive terms differ by a constant multiplier.
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Common RatioThe constant multiplier for a geometric sequence is called the common ratio, r.
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State whether the sequence 8, 4, 2, 1, … is arithmetic or geometric.
geometric
Example
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State whether the sequence –6, –18, –54, –162, … is arithmetic or geometric.
geometric
Example
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State whether the sequence 5, 7, 9, 11, … is arithmetic or geometric.
arithmetic
Example
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State whether the sequence 5, 10, 20, 40, … is arithmetic or geometric.
geometric
Example
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Geometric SequenceTerms differ by a constant factor r.
an = an – 1r
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Write the first six terms of the geometric sequence in which a1 = 1 and r = 3.a1 = 1a2 = 1 • 3 = 3a3 = 3 • 3 = 9a4 = 9 • 3 = 27
Example 1
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The first six terms of the sequence are
1, 3, 9, 27, 81, and 243.
Write the first six terms of the geometric sequence in which a1 = 1 and r = 3.
Example 1
a5 = 27 • 3 = 81a6 = 81 • 3 = 243
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Find the value of a1 for the sequence 2, 6, 18, 54, 162, 486, …
a1 = 2
Example 2
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Find the value of r for the sequence 2, 6, 18, 54, 162, 486, …
r = 3
Example 2
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Find the value of a3 for the sequence 2, 6, 18, 54, 162, 486, …
a3 = 18
Example 2
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Find the value of a8 for the sequence 2, 6, 18, 54, 162, 486, …
= 4,374
a7 = 486 • 3= 1,458
a8 = 1,458 • 3
Example 2
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Geometric SequenceTerms differ by a constant factor r.
an = an – 1r
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Write the first five terms of the sequence defined bya1 = –4 and an = 3an – 1.a1 = –4a2 = 3(–4) = –12a3 = 3(–12) = –36a4 = 3(–36) = –108a5 = 3(–108) = –324
Example 3
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Write the first five terms of the sequence defined bya1 = –4 and an = 3an – 1.
The first five terms of the sequence are
–4, –12, –36, –108, and –324.
Example 3
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Write the first four terms of the sequence defined bya1 = 2 and an = 4an – 1.
2, 8, 32, 128
Example
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Write the first four terms of the sequence defined bya1 = –3 and an = 2an – 1.
–3, –6, –12, –24
Example
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Find the common ratio, r, of the sequence 4, –12, 36, –108.
r = –3
Example
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Find the common ratio, r, of the sequence 24, 12, 6, 3.
r =
12
Example
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Write the recursive formula for the sequence 729, 243, 81, 27, ...a1 = 729r =
13
an = an – 113
Example 4
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3, 6 ,12, 24, 48, 96× 2 to get next term
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1Position Term
23456n
33 • 21 = 63 • 22 = 123 • 23 = 243 • 24 = 483 • 25 = 963 • 2n – 1
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Explicit FormulaThe explicit formula for a geometric sequence is an = a1r
n –1, in which a1 is the
first term and r is the common ratio.
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Write the explicit formula for the sequence –5, –15, –45, –135, –405, ...a1 = –5r = 3 an = –5(3)n – 1
Example 5
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Write the explicit formula for the sequence –3, –6, –12, –24, ...
an = –3(2)n – 1
Example
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Write the explicit formula for the sequence 12, 6, 3, 1.5, ...
an = 12( )n – 112
Example
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A ball bounces three-fourths the height of its fall. If the ball falls 12 ft., how high does it bounce on the first bounce? on the second bounce? on the third bounce?
9 ft.; 6.75 ft.; 5.0625 ft.
Exercise
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In the last problem, the height of the bounces forms a geometric sequence. Find the common ratio of this geometric sequence.
r = 0.75
Exercise
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If the ball falls 12 ft. and begins bouncing, what is the total distance it has traveled when it hits the ground the third time?
43.5 ft.
Exercise
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When will the ball stop bouncing?
Exercise