8.2 Arithmetic Sequences and Series
8.3 Geometric Sequences and Series
Arithmetic Sequences
A sequence is arithmetic if the differences
between consecutive terms are the same.
So, the sequence
a1, a2, a3, ….,an
is arithmetic if there is a number d such that
a2-a1= a3-a2=a4-a3=…=d
The number d is the common difference of
the sequence.
Examples
1) 5,8,11,14,17,...
2) 9,5,1,3,7,...
7 4 3 5
3) 1, , , , ,...
6 3 2 3
nth Term of an Arithmetic Sequence
T he nth term of an arithmetic sequence is
a n  a1  (n  1)d
where a1 is the first term of the sequence and d is the common difference .
Examples :
1) 5,8,11,14,17,...
2) 9,5,1,3,7,...
7 4 3 5
3) 1, , , , ,...
6 3 2 3
More examples
1)
The fourth term of an arithmetic sequence is 20, and the
13th term is 65. What is the nth term of the sequence?
2)
Find the tenth term of the arithmetic sequence that
begins with 8 and 20.
Sum of a Finite Arithmetic Sequence
The sum of a finite arithmetic sequence with n terms is given by
n
S n  a1  an 
2
Examples :
1) 40  37  34  31  28  25  22
2) Sum of the integers from 1 to 57
10
3)
 3n  2
n 1
25
3)
 6  4n
n 0
Exercises:
1)
Determine the seating capacity of an auditorium with 30
rows of seats if there are 20 seats in the first row, 22
seats in the second row, 24 in the third row and so on.
2)
Can you find the sum of an infinite arithmetic series?
Geometric Sequences
A sequence is geometric if the ratios of
consecutive terms are the same. So, the
sequence
a1, a2, a3, ….,an
is geometric if there is a number r such that
a2/a1= a3/a2=a4/a3=…=r
The number r is the common ratio of the
sequence.
Examples
1) 3,9,37,81,243,....
2) 10,20,40,80,160,...
1 1
1 1
1
3)  , , ,
,
,...
4 16 64 256 1024
nth Term of an Geometric Sequence
The nth term of an geometric sequence is
a n  a1r n 1
where a1 is the first term of the sequence and r is the common ratio.
Examples
1) 3,9,37,81,243,....
2) 10,20,40,80,160,...
1 1
1 1
1
3)  , , ,
,
,...
4 16 64 256 1024
More examples
1)
Find the nth term of a geometric sequence whose first
term is 4 and whose common ratio is ½ .
2)
The second term of a geometric sequence is -18 and the
fifth term is 2/3. Find the sixth term.
The Sum of a Finite Geometric Sequence
n


1

r
i 1

a1r  a1 

1

r
i 1


n

Examples :
7
1)
2
n 1
n 1
1
2)
5 
n 1  2 
5

n
Sum of an Infinite Geometric Series

a r
i 1
1
i 1
a1

1 r
r 1
Examples :

1)
2
n 1
n 1

1
2)
5 
n 1  2 

n
More examples
1) Find the sum 5 + .5
+ .05 + .005 + .0005
2) .

1
16 
 3
n 0

n