Transcript Sullivan College Algebra Section 9.3

Sullivan Algebra and
Trigonometry: Section 13.3
Objectives of this Section
• Determine if a Sequence Is Geometric
• Find a Formula for a Geometric Sequence
• Find the Sum of a Geometric Sequence
• Solve Annuity Problems
A sequence is geometric when the
ratio of successive terms is always
the same nonzero number. A
geometric sequence is defined
recursively as
a1  a
an 1  ran
where r is called the common ratio
of the sequence.
Determine if the following sequence is
geometric
3, 15, 75, 375, . . .
15
5
3
75
5
15
375
5
75
The sequence is geometric with a
common ratio of 5.
nth Term of a Geometric Sequence
For a geometric sequence an whose
first term a and whose common ratio
is r , the nth term is determined
by the formula
an  ar
n 1
, r0
Find the 11th term of the geometric sequence
1
4, 2, 1, ,
2
First term: 4
an  ar
n 1
1

a11  4 
 2
1
Common ratio:
2
1

 4 
 2
111
n 1
1

 4 
 2
10
1

256
Sum of First n Terms of a Geometric Sequence
Let an  be a geometric sequence with first term
a and common ratio r . The sum Sn of the first n
terms of an  is
1 rn
Sn  a
, r  0, 1
1 r
Find the sum of the first 10 terms
 1 n 
of the sequence   . That is, find


5


1 1
1



5 25 125
1

 
 5
10
Geometric with a  15 , r  15
10
10
1
1
1

1

1
1
5
5


S10  
4
5
5 1 1
5
5
 0.249999974
 
 
An infinite sum of the form
n 1
a  ar  ar   ar 
with first term a and common ratio r , is called
an infinite geometric series and is denoted by
2

 ar
k 1
k 1
If r  1, the sum of the infinite geometric

series  ar
k 1
k 1
is

 ar
k 1
k 1
a

1 r
6 12
Find the sum of 3   
5 25
a  3,
r 1
6
6 2
5
r


3 15 5
6 12
3  
5 25

3
2
1
5
5
Amount of an Annuity
If P represents the deposit made in dollars
at each payment period for an annuity at i
percent interest per payment period, the
amount A of the annuity after n payment
periods is
1  i  1

A P
n
i
Suppose Yola deposits $500 into a Roth IRA
every quarter (3 months). What will be the
value of the account in 25 years assuming it
earns 9% per annum compounded quarterly?
0.09
i
 0.0225, n  25(4)  100, P  $500
4
1  i  1
1  0.0225  1


A P
 $500
n
i
100
0.0225
 $183,423.25