Transcript Sec 11.3 Geometric Sequences and Series

Sec 11.3
Geometric Sequences and Series
Objectives:
•To define geometric sequences and
series.
•To define infinite series.
•To understand the formulas for sums
of finite and infinite geometric series.
An arithmetic sequence is defined when we
repeatedly add a number, d, to an initial term.
A geometric sequence is generated when we
start with a number, a1 , and repeatedly
multiply that number by a fixed nonzero
constant, r, called the common ratio.
Definition of a Geometric Sequence
A geometric sequence is a sequence of the form
a, ar, ar2, ar3, ar4, . . The number a is the first
term, and r is the common ratio of the
sequence.
The nth term of a geometric sequence is given by
an  ar
n 1
Ex. 1 Find the first four terms and the nth
term of the geometric sequence with a = 2
and r = 3.
Since a = 2 and r = 3, we can plug into the
nth term formula to get the terms.
an  ar
n 1
a1  2(3)  2
0
a2  2(3)  6
1
a3  2(3)  18
2
a4  2(3)  54
3
Ex 2. Find the eighth term of the geometric
sequence 5, 15, 45,…
an  ar
n 1
a8  5(3)
7
 5(2187)
 10,935
Ex 3. The third term of a geometric sequence is
63/4, and the sixth term is 1701/32. Find the
fifth term.
If we divide the equations we get the following:
1701
a6 
32
63
a3 
4
Using the nth term
formula and the two
given terms we get a
system of equations.
1701
 ar 5
32
63
 ar 2
4
1701
32  r 3
63
4
And so, we get:
27
r 
8
3
r
2
3
Since we have the sixth term, just divide it by r and
you will get the fifth term.
6
a
a5 
r
1701
567
32
a5 

3
16
2
Partial Sums
For the geometric sequence
a, ar, ar2, ar3, ar4, . . . , arn–1, . . . ,
the nth partial sum is:
 1 r 
Sn  a 

 1 r 
n
Ex 4. Find the sum of the first five terms of the
geometric sequence 1, 0.7, 0.49, 0.343, . . .
 1 rn 
Sn  a 

 1 r 
5
 1   0.7  

S5  1
 1  0.7 


S5  2.7731
Ex. 5 Find the sum.
5
 7 
k 1
2
3
k
 1 rn 
Sn  a 

 1 r 
  2 5 
 1    
14   3  
S5  
3
 2 
 1   3  



Plug 1 in for k and
you will get a = -14/3.
Then plug in a and r
into the sum formula.
32 

1


14
243
S5   

3 5 
 3 
 275 
14  243 
 

3 5 
 3 
770

243
You do not want to use
decimals. All answers will
be exact (which means they
will be given as fractions.)
We do not want to get
rounded answers.
Use the fraction key on your
calculator to help you get the
numbers you see here.
HW #3 Finite Geometric Series
Wkst odds (even extra credit)
Infinite Series
An expression of the form
a1 + a2 + a3 + a4 + . . .
is called an infinite series.
Let’s take a look at the partial sums of
this series.
1 1 1 1
1
  
   n  
2 4 8 16
2
Sum of an Infinite Geometric Series
If | r | < 1, the infinite geometric series
a + ar + ar2 + ar3 + ar4 + . . . + arn–1 + . . .
has the sum
a
S
1 r
Ex 6 Find the sum of the infinite geometric
2 2
2
2
series 2  5  25  125      5n    
HW #4 Infinite Series Wkst odds
(evens extra credit)