Finding the Common RatioDetermine the common ratio rfor the geometric sequence.
To find r, choose any two successive terms and divide the second one by the first.
We choose the second and third terms of the sequence.
Substitute.Write as division.Definition of divisionMultiply. Write in lowest terms.Any other two successive terms could have been used to find r. Additional terms of the
sequence can be found by multiplying each successive term by NOW TRY
OBJECTIVE 2 Find the general term of a geometric sequence. The general
term of a geometric sequence is expressed in terms of and rby
writing the first few terms as
which suggests the next rule.
a 1 , a 2 = a 1 r, a 3 =a 1 r^2 , a 4 = a 1 r^3 ,Á,
an a 1 , a 2 , a 3 ,Á a 1
12.
=
1
2
=
15
4
#^2
15
=
15
4
,
15
2
=
15
4
15
2r=
a 3
a 2
15,
15
2
,
15
4
,
15
8
,Á
EXAMPLE 1
692 CHAPTER 12 Sequences and Series
NOW TRY
EXERCISE 1
Determine rfor the geometric
sequence.
1
4, -1, 4, -16, 64,Á
NOW TRY ANSWERS
- 4
General Term of a Geometric SequenceThe general term of the geometric sequence with first term and common
ratio ris
ana 1 rn^1.
a 1
CAUTION In finding be careful to use the correct order of operations.
The value of rn-^1 must be found first. Then multiply the result by a 1.
a 1 rn-^1 ,
Finding the General Term of a Geometric SequenceDetermine the general term of the sequence in Example 1.
The first term is and the common ratio is
Substitute into the formula forIt is not possible to simplify further, because the exponent must be applied before the
multiplication can be done. NOW TRY
an= a 1 rn-^1 = 15 a an.
1
2
b
n- 1r=
1a 1 = 15 2.
NOW TRY EXAMPLE 2
EXERCISE 2
Determine the general term of
the sequence.
1
4, -1, 4, -16, 64,Á
- an=^14 1 - 42 n-^1