Thip k
How do you find
the common ratio
between two
adjacent terms?
Divide each term by the
previous term.
jy s u ^ ^ _
W hat do you need
to know in order to
w rite recursive and
explicit formulas for a
geometric sequence?
You need th e com m on
ratio and the starting
value.Identifying Geomet ric Sequences
Which of the following are geometric sequences?
0 20 200 2, 0 0 0 20,000 200 , 000 ,...X 10 X 10 X 10 X 10
There is a common ratio, r = 10. So, the sequence is geometric.0 2 4 6 8 10,...X 2 X 1.5 X 1.33 X 1.25
There is no common ratio. So, the sequence is not geometric.0 5 - 5 5 - 5 5,...x -1 x -1 x -1 x -1
There is a common ratio, r = -1. So, the sequence is geometric.vj Go t I t? 1. W hich of th e following are geom etric sequences? If the sequence is not
geom etric, is it arithm etic?
a. 3,6,12,24,48,... b. 3, 6 , 9 , 1 2 , 1 5 ,...
C. ^ • d. 4,7, 11,16,22,...Any geom etric seq u en ce can be w ritten w ith b o th an explicit a n d a recursive formula.
The recursive form ula is useful for finding th e next term in the sequence. The explicit
form ula is m ore convenient w h en finding th e n th term.Fi n d i n g Recu r si v e a n d Ex p l i ci t Fo r m u l as
Find the recursive and explicit formulas for the sequence 7,21,63,189,...The starting value al is 7. The com m on ratio r is — =^21 3.= a;an = • r Use the form ula. an = ax • r n~ x
aj = 7; an = a n_ 1 • r Substitute the starting value for av a n = 7 • r n~ l
ax = 7; • 3 Substitute the common ratio for r. a n = 7 • 3 ” _1The recursive formula is The explicit formula i
al = 7;an = an_1 *3. an = 7 • 3 n_1.w Got It? 2. Find th e recursive a n d explicit form ulas for each of th e following.
a. 2,4, 8 ,1 6 ,... b. 40,20,10, 5 ,...468 Chapter 7 Ex p o n e n t s a n d Ex p o n e n t i a l Fu n c t i o n s