College Algebra

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• College Algebra
• Fifth Edition
• James Stewart ? Lothar Redlin ? Saleem Watson

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• Sequences and Series

9
3

• Geometric Sequences

9.3
4
Introduction

• Geometric sequences occur frequently in
  applications to fields such as
• Finance
• Population growth

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• Geometric Sequences

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Arithmetic Sequence vs. Geometric Sequence

• An arithmetic sequence is generated when
• We repeatedly add a number d to an initial term
  a.
• A geometric sequence is generated when
• We start with a number a and repeatedly multiply
  by a fixed nonzero constant r.

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Geometric SequenceDefinition

• A geometric sequence is a sequence of the form
• a, ar, ar2, ar3, ar4, . . .
• The number a is the first term.
• r is the common ratio of the sequence.
• The nth term of a geometric sequence is given
  by an arn1

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Common Ratio

• The number r is called the common ratio because
• The ratio of any two consecutive terms of the
  sequence is r.

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E.g. 1Geometric Sequences
Example (a)

• If a 3 and r 2, then we have the geometric
  sequence
• 3, 3 2, 3 22, 3 23, 3 24, . . .
• or 3, 6, 12, 24, 48, . . .
• Notice that the ratio of any two consecutive
  terms is r 2.
• The nth term is an 3(2)n1

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E.g. 1Geometric Sequences
Example (b)

• The sequence
• 2, 10, 50, 250, 1250, . . .
• is a geometric sequence with a 2 and r 5
• When r is negative, the terms of the sequence
  alternate in sign.
• The nth term is an 2(5)n1.

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E.g. 1Geometric Sequences
Example (c)

• The sequence
• is a geometric sequence with a 1 and r ?
• The nth term is an (?)n1

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E.g. 1Geometric Sequences
Example (d)

• Heres the graph of the geometric sequence an
  (1/5) 2n 1
• Notice that the points in the graph lie on the
  graph of the exponential function y (1/5)
  2x1

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E.g. 1Geometric Sequences
Example (d)

• If 0 lt r lt 1, then the terms of the geometric
  sequence arn1 decrease.
• However, if r gt 1, then the terms increase.
• What happens if r 1?

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Geometric Sequences in Nature

• Geometric sequences occur naturally.
• Here is a simple example.
• Suppose a ball has elasticity such that, when it
  is dropped, it bounces up one-third of the
  distance it has fallen.

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Geometric Sequences in Nature

• If the ball is dropped from a height of 2 m, it
  bounces up to a height of 2(?) ? m.
• On its second bounce, it returns to a height of
  (?)(?) (2/9)m, and so on.

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Geometric Sequences in Nature

• Thus, the height hn that the ball reaches on its
  nth bounce is given by the geometric sequence
• hn ?(?)n1 2(?)n
• We can find the nth term of a geometric sequence
  if we know any two termsas the following
  examples show.

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E.g. 2Finding Terms of a Geometric Sequence

• Find the eighth term of the geometric sequence
  5, 15, 45, . . . .
• To find a formula for the nth term of this
  sequence, we need to find a and r.
• Clearly, a 5.

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E.g. 2Finding Terms of a Geometric Sequence

• To find r, we find the ratio of any two
  consecutive terms.
• For instance, r (45/15) 3
• Thus, an 5(3)n1
• The eighth term is a8 5(3)81 5(3)7
  10,935

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E.g. 3Finding Terms of a Geometric Sequence

• The third term of a geometric sequence is 63/4,
  and the sixth term is 1701/32.
• Find the fifth term.
• Since this sequence is geometric, its nth term
  is given by the formula an arn1.
• Thus, a3 ar31 ar2 a6 ar61 ar5

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E.g. 3Finding Terms of a Geometric Sequence

• From the values we are given for those two terms,
  we get this system of equations
• We solve this by dividing

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E.g. 3Finding Terms of a Geometric Sequence

• Substituting for r in the first equation, 63/4
  ar2, gives
• It follows that the nth term of this sequence
  is an 7(3/2)n1
• Thus, the fifth term is

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• Partial Sums of Geometric Sequences

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Partial Sums of Geometric Sequences

• For the geometric sequence a, ar, ar2, ar3,
  ar4, . . . , arn1, . . . , the nth partial sum
  is

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Partial Sums of Geometric Sequences

• To find a formula for Sn, we multiply Sn by r and
  subtract from Sn

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Partial Sums of Geometric Sequences

• So,
• We summarize this result as follows.

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Partial Sums of a Geometric Sequence

• For the geometric sequence an arn1, the nth
  partial sum Sn a ar ar2 ar3 ar4 .
  . . arn1 (r ? 1) is given by

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E.g. 4Finding a Partial Sum of a Geometric
Sequence

• Find the sum of the first five terms of the
  geometric sequence 1, 0.7, 0.49, 0.343, . . .
• The required sum is the sum of the first five
  terms of a geometric sequence with a 1 and
  r 0.7

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E.g. 4Finding a Partial Sum of a Geometric
Sequence

• Using the formula for Sn with n 5, we get
• The sum of the first five terms of the sequence
  is 2.7731.

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E.g. 5Finding a Partial Sum of a Geometric
Sequence

• Find the sum
• The sum is the fifth partial sum of a geometric
  sequence with first term a 7(?) 14/3 and
  common ratio r ?.
• Thus, by the formula for Sn, we have

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• What Is an Infinite Series?

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Infinite Series

• An expression of the form a1 a2 a3 a4
  . . . is called an infinite series.
• The dots mean that we are to continue the
  addition indefinitely.

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Infinite Series

• What meaning can we attach to the sum of
  infinitely many numbers?
• It seems at first that it is not possible to add
  infinitely many numbers and arrive at a finite
  number.
• However, consider the following problem.

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Infinite Series

• You have a cake and you want to eat it by
• First eating half the cake.
• Then eating half of what remains.
• Then again eating half of what remains.

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Infinite Series

• This process can continue indefinitely because,
  at each stage, some of the cake remains.
• Does this mean that its impossible to eat all
  of the cake?
• Of course not.

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Infinite Series

• Lets write down what you have eaten from this
  cake
• This is an infinite series.
• We note two things about it.

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Infinite Series

1. Its clear that, no matter how many terms of this
   series we add, the total will never exceed 1.
2. The more terms of this series we add, the closer
   the sum is to 1.

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Infinite Series

• This suggests that the number 1 can be written as
  the sum of infinitely many smaller numbers

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Infinite Series

• To make this more precise, lets look at the
  partial sums of this series

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Infinite Series

• In general (see Example 5 of Section 9.1),
• As n gets larger and larger, we are adding more
  and more of the terms of this series.
• Intuitively, as n gets larger, Sn gets closer to
  the sum of the series.

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Infinite Series

• Now, notice that, as n gets large, (1/2n) gets
  closer and closer to 0.
• Thus, Sn gets close to 1 0 1.
• Using the notation of Section 4.6, we can write
  
• Sn ? 1 as n ? 8

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Sum of Infinite Series

• In general, if Sn gets close to a finite number S
  as n gets large, we say that
• S is the sum of the infinite series.

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• Infinite Geometric Series

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Infinite Geometric Series

• An infinite geometric series is a series of the
  form
• a ar ar2 ar3 ar4 . . . arn1 . . .
• We can apply the reasoning used earlier to find
  the sum of an infinite geometric series.

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Infinite Geometric Series

• The nth partial sum of such a series is given
  by
• It can be shown that, if r lt 1, rn gets close
  to 0 as n gets large.
• You can easily convince yourself of this using a
  calculator.

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Infinite Geometric Series

• It follows that Sn gets close to a/(1 r) as n
  gets large, or
• Thus, the sum of this infinite geometric series
  is a/(1 r)

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Sum of an Infinite Geometric Series

• If r lt 1, the infinite geometric series
  a ar ar2 ar3 ar4 . . . arn1 . . .
  has the sum

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E.g. 6Finding the Sum of an Infinite Geometric
Series

• Find the sum of the infinite geometric series
• We use the formula.
• Here, a 2 and r (1/5).
• So, the sum of this infinite series is

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E.g. 7Writing a Repeated Decimal as a Fraction

• Find the fraction that represents the rational
  number .
• This repeating decimal can be written as a
  series

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E.g. 7Writing a Repeated Decimal as a Fraction

• After the first term, the terms of the series
  form an infinite geometric series with

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E.g. 7Writing a Repeated Decimal as a Fraction

• So, the sum of this part of the series is
• Thus,