10-7 Geometric Sequences 10-7 Geometric Sequences

10-7
Geometric Sequences
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Objectives
• Students will recognize and extend
geometric sequences.
• Students will find geometric means.
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Vocabulary
• geometric sequence – a sequence in which
each term after the nonzero first term is found by
multiplying the previous term by a constant
called the common ratio, where r ≠ 0, 1.
• common ratio – the number each term is
multiplied by to find the next term.
• geometric means – the missing term between 2
given terms in a geometric sequence. To find
the missing term, multiply the two terms that the
missing number is between and take the square
root.
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Formula for the nth Term of a
Geometric Sequence
a n = a 1 •r n-1
a n = nth term
a 1 = 1 st term
r = common ratio
n = term number
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Determine whether the sequence is geometric.
1, 4, 16, 64, 256, …
Determine the pattern.
1 4 16 64 256
In this sequence, each term in found by multiplying the
previous term by 4.
Answer: This sequence is geometric.
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Determine whether the sequence is geometric.
1, 3, 5, 7, 9, 11, …
Determine the pattern.
1 3 5 7 9 11
In this sequence, each term is found by adding 2 to the
previous term.
Answer: This sequence is arithmetic, not geometric.
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Determine whether each sequence is geometric.
a.
Answer: yes
b. 0, 11, 22, 33, 44, …
Answer: no
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Find the next three terms in the geometric sequence.
20, –28, 39.2, …
Divide the second term by the first.
The common factor is –1.4. Use this information to find the
next three terms.
20, –28, 39.2
–54.88 76.832 –107.5648
Answer: The next three terms are –54.88,
76.832, and –107.5648.
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Find the next three terms in the geometric sequence.
64, 48, 36, …
Divide the second term by the first.
The common factor is 0.8. Use this information to find the
next three terms.
64, 48, 36
27 20.25 15.1875
Answer: The next three terms are 27, 20.25,
and 15.1875.
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Find the next three terms in each geometric sequence.
a. 42, 25.2, 15.12, …
Answer: 9.072, 5.4432, 3.26592
b. 100, –250, 625, …
Answer: –1562.5, 3906.25, –9765.625
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Geography The population of the African country of
Liberia was about 2,900,000 in 1999. If the population
grows at a rate of about 5% per year, what will the
population be in the years 2003, 2004, and 2005?
The population is a geometric sequence in which the first
term is 2,900,000 and the common ratio is 1.05.
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Year
1999
2000
2001
2002
2003
2004
2005
Population
2,900,000
2,900,000(1.05) or 3,045,000
3,045,000(1.05) or 3,197,250
3,197,250(1.05) or 3,357,112.5
3,357,112.5(1.05) or 3,524,968.1
3,524,968.1(1.05) or 3,701,216.5
3,701,216.5(1.05) or 3,886,277.3
Answer: The population of Liberia in the years 2003,
2004, and 2005 will be about 3,524,968, 3,701,217,
and 3,886,277, respectively.
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Geography The population of the Baltic State of
Latvia was about 2,500,000 in 1998. If the population
grows at a rate of about 2% per year, what will the
population be in the years 2003 and 2004?
Answer: about 2,760,202 in 2003 and about 2,815,406
in 2004
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Find the eighth term of a geometric sequence in
which
Formula for the nth term of
a geometric sequence
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Answer: The eighth term in the sequence is 15,309.
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Find the ninth term of a geometric sequence in
which
Answer: 786,432
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Find the geometric mean in the sequence 7, ___, 112.
In the sequence, and To find
you must first find r.
Formula for the nth term of
a geometric sequence
Divide each side by 7.
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Simplify.
Take the square root of each side.
If the geometric mean is 7(4) or 28. If
the geometric mean is 7(–4) or –28.
Answer: The geometric mean is 28 or –28.
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Find the geometric mean in the sequence 9, ___, 576.
Answer: 72 or –72
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