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10.3 Geometric Sequences

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					10.3 Geometric Sequences
10.3 Geometric Sequences
and Series
   If we start with a number, a1, and
    repeatedly multiply it by some constant, r,
    then we have a geometric sequence:
              a1, a1r, a1r 2, a1r 3, a1r 4,….
       The nth term of a geometric sequence is given
                    n 1
        by an  a1r , for any integer n  1

       The number r is called the common ratio
10.3 Geometric Sequences
and Series (Example 2)
 Find the 8th term of the geometric
 sequence: 5, 15, 45, …

Solution:
Use formula, an = a1r (n – 1)
a1 =        r=          n=


(Now work Example 3 in text….)
Geometric Series
   The sum of the terms of a geometric sequence is
    called a geometric series.
   For example:
       S n  a  ar  ar 2  ar 3  ....  ar n  2  ar n 1

    is a finite geometric series with common ratio r.

   What is the sum of the n terms of a finite geometric
    series?
Deriving a formula for the nth
partial sum of a geometric series
        S n  a  ar  ar 2  ....  ar n  2  ar n 1
       rS n      ar  ar 2 + .... + ar n  2  ar n 1  ar n

 S n  rS n  a  0  0  . . . . .  0  0              ar n

S n (1  r )  a(1  r n )

              (1  r n )
       Sn  a
               (1  r )
10.3 Geometric Sequences
and Series
   The sum of the terms of an infinite
    geometric sequence is an infinite
    geometric series.
       For some geometric sequences, Sn gets close
        to a specific number as n gets large.
       This number becomes the limit of the sum of
        the infinite geometric sequence.
       When |r|<1, the limit or sum of an infinite
        geometric series is given by S  a1 .
                                        1 r
10.3 Geometric Sequences
and Series
   You should be able to:
       Identify the common ratio of a geometric
        sequence, and find a given term and the
        sum of the first n terms.
       Find the sum of an infinite geometric
        series, if it exists.
10.1 Sequences and Series
   You should be able to:
       Find terms of sequences given the nth
        term.
       Look for a pattern in a sequence and try to
        determine a general term.
       Convert between sigma notation and other
        notation for a series.
       Construct the terms of a recursively
        defined sequence.

				
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