Geometric sequence

2. Geometric Sequences and Series
A sequence is geometric if the ratios of consecutive
terms are the same.
2, 8, 32, 128, 512, . . .
geometric sequence
The common ratio, r, is 4.
8
2
4

32
8
4

128
32
4

512
128
4


3. Example 1.
a. Is the sequence geometric? If so, what is ?
2,4,8,16,...2 ,...
n
r
4 8 16
2, 2, 2
2 4 8
   2
r 
b. Is the sequence geometric? If so, what is ?
1 1 1 1 1
, , , ,..., ,...
3 9 27 81 3
n
r
 
  
 
 
1 1 1
1 1 1
9 27 81
1 1 1
3 3 3
3 9 27

     
 
1
3
r  

4. The nth term of a geometric sequence has the form
an = a1rn - 1
where r is the common ratio of consecutive terms of
the sequence.
15, 75, 375, 1875, . . .
a1 = 15
The nth term is 15(5n-1).
75 5
15
r  
a2 =
15(5)
a3 =
15(52)
a4 =
15(53)

5. 1
Example 2. Write the first five terms of the geometric sequence whose first term is a 3
and whose common ratio is 2.
r


1
1
2
3
3 2 6
a
a

  
2
3 3 2 12
a   
3
4 3 2 24
a   
4
5 3 2 48
a   

6. Example 3. Find the 15th term of the geometric sequence whose first term
is 20 and whose common ration is 1.05.
1
1
n
n
a a r 

 
14
15 20 1.05
a  39.599


7. Example 4. Find a formula for the nth term of the following geometric
sequence. What is the ninth term of the sequence?
5, 15, 45, …
Find the common ratio
15 5 3 45 15 3
   
 
1
5 3
n
n
a


 
8
9 5 3
a  32,805


8. 125
Example 5. The 4th term of a geometric sequence is 125, and the 10th term is .
64
Find the 14th term. Assume that the terms of the sequence are positive.
10 4
10 4
a a r 

6
125
125
64
r

6
1
64
r

1
2
r

4
14 10
a a r

4
125 1
64 2
 
  
 
125
1024


9. The sum of the first n terms of a sequence is
represented by summation notation.
1 2 3 4
1
n
i n
i
a a a a a a

     

index of
summation
upper limit of summation
lower limit of summation
5
1
4n
n

 1 2 3 4 5
4 4 4 4 4
   
4 16 64 256 1024
    
1364


10. The sum of a finite geometric sequence is given by
 
1
1 1
1
1 .
1
n n
i
n
i
r
S a r a
r



 


5 + 10 + 20 + 40 + 80 + 160 + 320 + 640 = ?
n = 8
a1 = 5
   
1
8
1 1
1
2
2
1
5
n
n
r
S a
r
 
 
 
5
2
10
r  
 
1 256
5
1 2


  
255
5
1



1275


11.  
12
1
Example 6. Find the sum 4 0.3
n
n

Write out a few terms.
         
12
1 2 3 12
1
4 0.3 4 0.3 4 0.3 4 0.3 ... 4 0.3
n
n
    

 
1 4 0.3 0.3 and 12
a r n
   
 
12
1
1
1
4 0.3
1
n
n
n
r
a
r

 

  

 

 
 
12
1 0.3
4 0.3
1 0.3
 

  

 
 
1.714

If the index began at i = 0, you would have to adjust your formula
     
12 12
0
0 1
4 0.3 4 0.3 4 0.3
n
n
i n
 
 
   
12
1
4 4 0.3
n
n
   4 1.714 5.714
  

12. The sum of the terms of an infinite geometric
sequence is called a geometric series.
a1 + a1r + a1r2 + a1r3 + . . . + a1rn-1 + . . .
If |r| < 1, then the infinite geometric series
1
1
0
.
1
i
i
a
S a r
r


 


has the sum
If 1, then the series does not have a sum.
r 

13. Example 7. Use a graphing calculator to find the first six partial sums of
the series. Then find the sum of the series.
 
1
cum sum(seq(4*0.6 , ,1,6))
x
x

4, 6.4, 7.84, 8.704, 9.2224, 9.53344
Use the formula for the sum of an infinite series to find the sum.
4
10
1 0.6
S  

 
1
1
4 0.6
n
n





14. Example 8. Find the sum of 3 + 0.3 + 0.03 + 0.003 + …,
3
3.33
1 0.1
S  


15. Example 9. A deposit of $50 is made on the first day of each month in a
savings account that pays 6% compounded monthly. What is the
balance at the end of 2 years?
This type of savings plan is called an increasing annuity.
The first deposit will gain interest for 24 months, and its balance will be
The second deposit will gain interest for 23 months
 
24
24
24
0.06
50 1 50 1.005
12
A
 
  
 
 
 
23
23
23
0.06
50 1 50 1.005
12
A
 
  
 
 
The last deposit will gain interest for only one month
 
1
1
0.06
50 1 50 1.005
12
A
 
  
 
 
The total balance will be the sum of the balances of the 24 deposits.
 
 
24
1
1 1.005
1
50 1.005 $1277.96
1 1 1.005
n
n
r
S a
r
 

 

  
 
 
 
   
 

16. Homework
Page 607-608
2-24 even, 25-31 odd, 33, 35, 40, 42, 48-54
even