5128_Ch08_434-471

Section 8.1 Sequences 437
(b) a 9 ln 2 (9 1)(ln 3) ln 2 8 ln 3 ln (2 • 3 8 ) ln 13,122
(c) The sequence is defined recursively by a 1 ln 2 and a n a n1 ln 3 for all n 2.
(d) The sequence is defined explicitly by a n ln 2 (n 1)(ln 3) ln (2 • 3 n1 ).
Now try Exercise 13.
DEFINITION
Geometric Sequence
A sequence {a n } is an geometric sequence if it can be written in the form
{a, a • r, a • r 2 , … , a • r n1 , …}
for some nonzero constant r. The number r is the common ratio.
Each term in a geometric sequence can be obtained recursively from its preceding
term by multiplying by r:
a n a n1 • r for all n 2.
EXAMPLE 4
Defining Geometric Sequences
For each of the following geometric sequences, find (a) the common ratio, (b) the tenth
term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term.
Sequence 1: 1, 2, 4, 8, 16, …
SOLUTION
Sequence 1
Sequence 2: 10 2 ,10 1 , 1, 10, 10 2 , …
(a) The ratio between successive terms is 2.
(b) a 10 (1) (2) 9 512
(c) The sequence is defined recursively by a 1 1 and a n (2)a n1 for n 2.
(d) The sequence is defined explicitly by a n (1) (2) n1 (2) n1 .
Sequence 2
(a) The ratio between successive terms is 10.
(b) a 10 (10 2 ) (10 9 ) 10 7
(c) The sequence is defined recursively by a 1 10 2 and a n (10)a n1 for n 2.
(d) The sequence is defined explicitly by a n (10 2 ) (10 n1 ) 10 n3 .
Now try Exercise 17.
EXAMPLE 5
Constructing a Sequence
The second and fifth terms of a geometric sequence are 6 and 48, respectively. Find
the first term, common ratio, and an explicit rule for the nth term.
SOLUTION
Because the sequence is geometric the second term is a 1 • r and the fifth term is a 1 • r 4 ,
where a 1 is the first term and r is the common ratio. Dividing, we have
a 1 •
a
r4
• r
4 8
6
1
r 3 8
r 2.
continued