Download Geometric Sequences

Math 30P
Geometric Sequences
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Believe it or not we studied sequences in grade 10
A sequence is a set of numbers in a definite
order which are written separated by commas
A common way to define a sequence
is in terms of the nth term formula
Ex #1 Write the next two terms of the following sequence and describe a rule which can be used to form the sequence
a) 1, 3, 5, 7, ...
c) 2, 3, 5, 7, 11, ...
b) 2, 4, 8, 16, ...
d) 1, 1, 2, 3 ,5, ...
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Some terminology
The first term of a sequence is written t1, the second term of a sequence is written as t2 and the general term, or the nth term is
written as tn. The general term provides a formula which can be used to determine any term of the sequence
Example: 1, 3, 5, 7, ... has the nth term formula tn= 2n ­ 1
Infinite Sequence is a seqeunce that goes on forever 3, 6, 12, ...
Finite sequence is one that has a specific number of terms 3, 6, 12, ... 192
The general term for the sequence 3, 6, 12,... is tn=3(2)n­1
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Ex #2
a) List the first three terms of the sequence tn = 3n2­n
b) Determine the 8th term of the seqeunce tn = 4n2 ­ 9
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Types of Sequences
Others
Arithmetic
­ terms are created by adding the same number to
the preceding term
1, 7, 13, 19, ...
­ they have a common difference
(6 above)
Geometric ­ terms are created by multiplying the same number to
the preceding term
2, 4, 8, 16, 32,...
­ Math 10P
­ They have a common ratio
(2 above)
­ tn = t1 + (n­1)d
­ Find common ration by
t2
t1
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Finding the Common ration
In order to find the common ratio we must divide two
adjacent terms. The term farther to the right in the sequence must
go in the numerator
Find the common ratio
a) 6, 12, 24, ....
b) ­1, 5, ­25, ...
c) ­10, ­5, ­5/2,...
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Developing the nth term formula
Consider the following sequence 3, 6, 12, 24, ...
t1 = 3
t2 = 6 = 3(2)1
t3 =
t4 =
t5 =
t6 =
The general formula for a geometric sequence
n­1
tn = ar
a = t1
r = common ratio
n = position of the term being considered
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Ex#5: Determine the general term (nth term formula) and calcualte the seventh for each of the following
a) 32, ­16 ,8, . . . b) 1/3 , ­1/6 , 1/12, .. . .
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Write the general term for the sequence 1/4, 1/8, 1/16, ... Given the sequence 32, 64, 128, ... , 16383, determine:
a) The number of terms in the sequence
b) Which term is 1024?
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Given the geometric sequence with t4=­54 and t7=1458, find the:
a) first term
b) common ratio
c) general term
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Ex #9
x, x + 5, x + 9 are the first three terms in a geometric sequence. Determine
the exact value of each term
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Geometric means
these are terms inserted between adjacent terms
Insert 4 geometric means between 81 and 1/729
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PAges 164­168 #1­15
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