# annuity_Thinking Mathematically by Robert Blitzer

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```					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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 views: 28 posted: 12/2/2010 language: English pages: 13