Arithmetic and Geometric Sequences 
 Sections 8.3/8.4  
Arithmetic Sequence:  is a sequence with a common
difference, a_{n}
= a_{1} +
(n 
1)d where d is the common difference. 
Geometric Sequence:  is a sequence with a common
ratio, a_{n}
= a_{1}r^{n
 1 } where r is the common ratio. 
Find the nth term of the following arithmetic sequence 5,
8, 11, 14, 17, ... Then find the 100th term. 
Solve 
Step 
Spot Check 
d = 8 
5 = 3,
d = 11  8 = 3 
Find
the common difference d = 3 
a_{1} = 3(1) + 2 = 5 
a_{n} = a_{1} + (n  1)d 
Use the formula for arithmetic sequence since there is a common difference  a_{3} = 3(3) + 2 = 9 + 2 = 11 
a_{1} = 5 and d = 3 
a_{1} = 5 is the first term of the sequence.  a_{5} = 3(5) + 2 = 15 + 2 = 17 
a_{n} = 5 + (n  1)3 
Plug in the values...  
a_{n} = 5 + 3n  3 = 3n + 2 
... and simplify.  
a_{n} = 3n + 2 
nth term  
a_{100} = 3(100) + 2 = 300 + 2 = 302 
100th term  

Find the nth term of the following arithmetic sequence 29,
25, 21, 17, 13, ... Then find the 25th term. 
[Solution] 


Find the sum of the first 20 terms of the arithmetic sequence 2, 6, 10,... . 
First find the nth term. Then use the formula for the sum of an arithmetic sequence.  
Solve  Step 
Spot Check 
d = 6  2 = 4, d = 10  6 = 4 
Find
the common difference d = 4 
a_{1} = 4(1)  2 = 2 
a_{n} = a_{1} + (n  1)d 
Use the formula for arithmetic sequence.  a_{3} = 4(3)  2 = 12  2 = 10 
a_{1} = 2 and d = 4 
a_{1} = 2 is the first term of the sequence.  
a_{n} = 2 + (n  1)4 
Plug in the values...  
a_{n} = 2 + 4n  4 = 4n  2 
... and simplify.  
a_{n} = 4n  2 
nth term  
Use the formula for the sum of an arithmetic sequence.  Alternate formula  
n = number of terms a = first term l = last term 

a = 2, l = a_{20} = 4(20)  2 = 80  2 = 78 
n = 20  
Plug in the values... and simplify.  

Find the sum of the first 30 terms of the arithmetic sequence 1, 5, 9,... . 
[Solution] 


Find the nth term of the following
geometric sequence 81, 27, 9, 3, 1, ... Then find the 7th term. 
Solve  Step 
Spot Check 
Find
the common ratio 

r = 1/3  
a_{n} = a_{1}r^{n  1 } 
Use the formula for geometric sequence.  
a_{1} = 81 and r = 1/3 
a_{1} = 81 is the first term of the sequence.  

nth term  
7th term  

Find the nth term of the following
geometric sequence 1, 5, 25, 125, 625, ... 
[Solution] 


Find the value of $10,000 left on deposit for 20 years at an annual rate of 12% compounded quarterly. 
This is a geometric sequence problem with a small change. a_{n} = a_{0}(1+ r)^{n }Let a_{0 }= initial value. 

Solve  Step  
a_{0 } = 10,000 
a_{0 }= initial value  
r = 0.12 ÷ 4 = 0.03  r = 12% annual rate divided by 4 (quarterly is 4 times a year)  
n = 20 × 4 = 80  n = number of years times by 4 (quarterly is 4 times a year)  
a_{80} = 10,000(1 + 0.03)^{80}^{ }= 

a_{80} = 10,000(1.03)^{80}^{ }» $106,408.91 

The account would have $106,408.91 at the end of 20 years. 


Find the value of $20,000 left on deposit for 30 years at an annual rate of 9% compounded monthly. 
[Solution] 


Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada