annuity_Thinking Mathematically by Robert Blitzer

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annuity_Thinking Mathematically by Robert Blitzer Powered By Docstoc
					Geometric
Sequences
     Definition of a Geometric
             Sequence
• A geometric sequence is a sequence in
  which each term after the first is obtained
  by multiplying the preceding term by a
  fixed nonzero constant. The amount by
  which we multiply each time is called the
  common ratio of the sequence.
                   The Common Ratio
The common ratio, r, is found by dividing any term after the first term by
the term that directly precedes it. In the following examples, the common
ratio is found by dividing the second term by the first term, a2  a1.

Geometric sequence                           Common ratio
1, 5, 25, 125, 625...                        r=51=5
6, -12, 24, -48, 96....                      r = -12  6 = -2
                        Text Example
Write the first six terms of the geometric sequence with first term 6 and common
ratio 1/3 .


Solution The first term is 6. The second term is 6       1/ , or 2. The third
                                                            3
term is 2  1/3, or 2/3. The fourth term is 2/3           1/ , or 2/ , and so
                                                             3        9
on. The first six terms are
                            6, 2, 2/3, 2/9, 2/27, 2/81.
   General Term of a Geometric
            Sequence
• The nth term (the general term) of a
  geometric sequence with the first term a1
  and common ratio r is
• an = a1 r n-1
                        Text Example
Find the eighth term of the geometric sequence whose first term is
-4 and whose common ratio is -2.


Solution To find the eighth term, a8, we replace n in the formula with 8, a1
with -4, and r with -2.
             an = a1r n - 1
             a8 = -4(-2)8 - 1 = -4(-2)7 = -4(-128) = 512

The eighth term is 512. We can check this result by writing the first eight terms
of the sequence:
                      -4, 8, -16, 32, -64, 128, -256, 512.
The Sum of the First n Terms of
    a Geometric Sequence
The sum, Sn, of the first n terms of a
geometric sequence is given by
              a1 (1 - r )
                       n
         Sn =
                 1- r
in which a1 is the first term and r is the
common ratio (r  1).
                   Example
• Find the sum of the first 12 terms of the
  geometric sequence:
  4, -12, 36, -108, ...
  Solution:
                    a1 (1 - r n )
              Sn =
                       1- r
                    a1 (1 - r 12 ) 4(1 - (-3)12 )
              S12 =               =
                        1- r          1 - (-3)
                4(1 - 531441)
              =                   = -531440
                        4
 Value of an Annuity: Interest
 Compounded n Times per Year
   If P is the deposit made at the end of each
compounding period for an annuity at r percent
 annual interest compounded n times per year,
    the value, A, of the annuity after t years:
                    r nt
                (1  ) - 1
           A= P     n
                     r
                     n
                      Example
• To save for retirement, you decide to deposit
  $2000 into an IRA at the end of each year for the
  next 30 years. If the interest rate is 9% per year
  compounded annually, find the value of the IRA
  after 30 years.
Solution:
                P=2000, r =.09, t = 30, n=1

                     r nt               .09 1*30
                 (1  ) - 1        (1      ) -1
            A= P     n      = 2000       1
                      r                   .09
                      n                    1
              Example cont.
• To save for retirement, you decide to deposit
  $2000 into an IRA at the end of each year for the
  next 30 years. If the interest rate is 9% per year
  compounded annually, find the value of the IRA
  after 30 years.      .09 1*30
                      (1    )   -1
Solution:    = 2000       1
                           .09
                            1
                    (1.09) 30 - 1
             = 2000
                         .09
                    12.2677
             = 2000            = $272,615.08
                       .09
The Sum of an Infinite Geometric
            Series
  If –1 < r < 1, then the sum of the infinite geometric
  series
                  a1+ a1r + a1r2 + a1r3 +…
in which a1is the first term and r is the common ratio
is given by
                         a1
                     S=
                        1- r
   If |r| >1, the infinite series does not have a sum.
Geometric
Sequences

				
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posted:12/2/2010
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