© 2008 Turner Educational Publishing
Math 242 Enrichment Problems Section 9.3
Iteration and Geometric Sequences
If we know any term in a geometric sequence, we can easily find the next term by multiplying by the
common ratio r. We can use this property of a geometric sequence to define the sequence recursively.
GEOMETRIC SEQUENCE – RECURSIVE FORMULA
A geometric sequence {an} with first term a and common ratio r can be defined recursively as
a1 = a; an+1 = r⋅an, n ≥ 1
Notice that the recursive formula an+1 = r⋅an describes a linear relationship, or direct variation,
between consecutive terms, which we can interpret in the context of iteration. If we define f(x) = r⋅x,
then the geometric sequence {an} is the orbit of the seed x0 = a1 under f. The graph of f is a line passing
through the origin with a slope of r. Figure 1 shows a cobweb diagram of the orbit. Because of the slope,
the points either get closer to one another or spread further apart. Every geometric sequence can be
interpreted as the orbit of some seed under a linear function passing through the origin with slope of r.
y
Figure 1
EXERISES
1. Consider the geometric sequence {an} = {4, 2, 1, 1
2, 1
4, …} from the class lecture.
(a) This sequence is the orbit of what seed under what function?
(b) Sketch a cobweb diagram for the first five terms of the sequence.
2. Consider the geometric sequence {an} = {3, −12, 48, −192, 768, …} from the class lecture.
(a) This sequence is the orbit of what seed under what function?
(b) Sketch a cobweb diagram for the first three terms of the sequence.
3. Let g(x) = bx for b > 0 and b ≠ 1. Since g is an exponential function, if we restrict the domain of g to
the natural numbers, we will obtain a geometric sequence.
(a) Write the first five terms of the geometric sequence generated by g.
(b) This sequence is the orbit of what seed under what function?
(c) Sketch a representative cobweb diagram for the first four terms of the sequence if b < 1.
(d) Sketch a representative cobweb diagram for the first four terms of the sequence if b > 1.
(e) Use your graphs from (c) and (d) to explain why exponential growth results when b > 1 but
exponential decay results when b < 1.
x
(a1, a2)
a1
(a2, a3)
(a3, a4) y = x
f(x) = r⋅x
a2 a3 a4
