8.3 Geometric Sequences · 2018. 11. 28. · The third term of a geometric sequence is , and the...
Transcript of 8.3 Geometric Sequences · 2018. 11. 28. · The third term of a geometric sequence is , and the...
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8.3 Geometric Sequences
DEFINITION OF A GEOMETRIC SEQUENCEA geometric sequence is a sequence of the form
The number a is the first term, and r is the common ratio of the sequence. the nth term of a geometric sequence is given by
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Write the first 5 terms and the nth term of the geometric sequence if a = 3 and r = 2
Find r, a and the nth term of the following sequence:
r = , a = 1, an = 1( )n 113
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Are the following sequences geometric? If so, find r and the nth term.
2,10, 50, 250, 1250,...
3, 6, 9, 12,...
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an = 4 + 3n an = (1)n 2n
an = nn
Are the following sequences geometric? If so, state the first 5 terms, the common ratio, & express the nth term in the form an = a(r)n - 1.
terms:
Not geometric
GeometricNot geometric
terms:terms:
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Find the 8th term of the geometric sequence5, 15, 45,...
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The third term of a geometric sequence is , and the sixth term is . Find the fifth term.
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The second and fifth terms of a geometric sequence are 30 & 3750, respectively. Which term of the sequence is 468,750?
468,750 is the 8th term.
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PARTIAL SUM OF A GEOMETRIC SEQUENCE
For the geometric sequence an=arn1, the nth partial sum
is given by
This is also called the sum of a finite geometric series.
(proof on page 610 in textbook)
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Find the sum of the first five terms of the geometric sequence
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Find the sums:1
38581
770243
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Using a picture and partial sums find
Partial sums: 164+
132+
14+
12 1
8+
116+
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if
then
We use this to find the actual sum of
If |r|<1, then the value of this term goes to zero as n goes to infinity.
So this is the formula for an infinite geometric series if |r|<1.
Example:
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SUM OF AN INFINITE GEOMETRIC SERIESIf |r|<1, then the infinite geometric series
converges and has a sum
If |r|>1, the series diverges.
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Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find the sum.
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Find the fraction that represents the rational numbers: 0.123123123...
0.123123123...
First write the repeating decimal as a sum:
Next write each decimal as a fraction:
Now plug "a" and "r" into the formula:
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Assignment:
Section 8.3 on Webassign