Module 23  Sequences and Series  
Introduction  Lesson 1  Lesson 2  Lesson 3  SelfTest  
Lesson 23.3: Recursively Defined Sequences  
This lesson investigates sequences that are defined recursively by examining the graph of the sequence and looking at the associated table of values. Defining a Recursive Sequence A recursive sequence defines a term a_{n} using one or more of the previous terms of the sequence. Because each term is defined using previous terms, the first few terms of the sequence must be defined explicitly. Explore the recursive sequence shown below that defines each term of the sequence as twice the previous term, a_{n} = 2a_{n  1}. The first term of the sequence is defined to be 1. The sequence may be written symbolically as two statements. a_{1} = 1 a_{n} = 2a_{n  1}, for n = 2, 3, 4, 5,... Graphing the Recursive Sequence Define the sequence in the Y= Editor.
The function notation u(n  1) represents the function value at n  1 and also represents the The first term of the sequence must also be entered so that the calculator will have a value with which to evaluate the second term. Subsequent terms will be determined by using previously computed terms.
u(nMin) represents u(1), the first term of the sequence.
The graph suggests that the sequence gets large without bound as n approaches infinity. If this is so, then the sequence diverges. Creating a Table of Values The divergence of the sequence can by supported by viewing a table of values.
It appears from the table that this sequence is the sequence a_{n} = 2^{n  1}. The sequence a_{n} = 2^{n  1} is an example of a geometric sequence. This particular geometric sequence is in fact divergent. Defining a Recursive Sequence Requiring Two Previous Values In the sequence investigated above, each term was defined as twice the value of the preceding term. Recursive sequences can also be defined using more than one preceding term. Explore the recursive sequence that defines each term as the sum of the two previous terms where a_{1} = a_{2} = 1 and a_{n} = a_{n1} + a_{n2} for n = 3, 4, 5,... .
The initial value u(nMin) is the list {1,1}, which represents the values of the first two terms. The symbols '{' and '}' are found above the parentheses keys.
23.3.1 Display the table and, if possible, identify the sequence shown. Although sequences can be defined recursively using more than the previous two terms, the TI83 is not able to do that. 

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