OBJECTIVE 3 Find any specified term of a geometric sequence.We can
use the formula for the general term to find any particular term.
Finding Specified Terms in SequencesEvaluate the indicated term for each geometric sequence.
(a) ;
Use the formula for the general term.
Formula for
Let
Let=- 972 Simplify.
= 4 # 1 - 325 a 1 =4, r=-3.
a 6 = a 1 #r^6 -^1 n=6.
an= a 1 rn-^1 an
a 1 = 4,r=- 3 a 6
EXAMPLE 3
SECTION 12.3 Geometric Sequences 693
NOW TRY
EXERCISE 3
Evaluate the indicated term
for each geometric sequence.
(a) ;
(b)10, 2, 52 , 252 ,Á; a 7
a 1 =3, r= 2 a 8NOW TRY ANSWERS
- (a)
(b) (^10) A^15 B^6 = 31252
31227 = 384
(b) ;
LetApply the exponent.Multiply. NOW TRYWriting the Terms of a SequenceWrite the first five terms of the geometric sequence whose first term is 5 and whose
common ratio is
a 5 = 5 a
1
2
b
4=
5
16
a 4 = 5 a
1
2
b
3=
5
8
,
a 3 = 5 a
1
2
b
2=
5
4
a 2 = 5 a ,
1
2
b =
5
2
a 1 = 5 , ,
12.
EXAMPLE 4
=
3
256
=
3
4
#^1
64
a 7 = a 1 = 43 ,r=^12 ,n=7.
3
4
a^1
2
b
7 - 1a 7
3
4
,
3
8
,
3
16
,Á
Use with
and
n=1, 2, 3, 4, 5.a 1 =5, r=^12 ,an=a 1 rn-^1 ,OBJECTIVE 4 Find the sum of a specified number of terms of a geometric
sequence.It is convenient to have a formula for the sum of the first nterms of a
geometric sequence. We can develop a formula by first writing out
Next, we multiply both sides by
Now add.
Factor on the left.Factor on the right.Sn Divide each side by 1-r.
a 111 rn 2
1 r
Sn 11 - r 2 =a 1 - a 1 rn
Sn-rSn=a 1 - a 1 rn
- rSn=-a 1 r- a 1 r^2 - a 1 r^3 - Á- a 1 rn-^1 - a 1 rn
Sn=a 1 + a 1 r+ a 1 r^2 +a 1 r^3 +Á+a 1 rn-^1
- rSn=-a 1 r- a 1 r^2 - a 1 r^3 - a 1 r^4 - Á- a 1 rn
- r.
Sn=a 1 + a 1 r +a 1 r^2 +a 1 r^3 +Á+ a 1 rn-^1
Sn.
Sn
NOW TRY
EXERCISE 4
Write the first five terms of
the geometric sequence whose
first term is 25 and whose
common ratio is -^15.
a 4 =- 51 ,a 5 = 251a 1 =25, a 2 =-5,a 3 =1,NOW TRYEvaluate and
then multiply.1 - 325