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The concept of geometric sequences and their recursive formulas. It explains how to define a geometric sequence recursively and interprets it as the orbit of a seed under a linear function. The document also includes exercises to practice identifying the seed and function for given geometric sequences and sketching cobweb diagrams.
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© 2008 Turner Educational Publishing
Math 242 Enrichment Problems Section 9.
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.
A geometric sequence { an } with first term a and common ratio r can be defined recursively as
a 1 = a ; an +1 = r ⋅ a (^) n , 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 x 0 = a 1 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
(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.
(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.
x
( a 1 , a 2 )
a 1
( a 2 , a 3 )
( a 3 , a 4 ) y = x
f ( x ) = r ⋅ x
a 2 a 3 a 4