Recursive Sequences
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Recursion is the process of choosing a starting term and repeatedly applying the same process to each term to arrive at the following term.  Recursion requires that you know the value of the term immediately before the term you are trying to find.

A recursive formula always has two parts:
  1.  the starting value for a1.

  2.  the recursion equation for an as a function of an-1
(the term before it.)
 

Recursive formula:
 

Same recursive formula:
   

Be sure you understand that the two formulas at the left say the same thing.  Different textbooks write recursive formulas in different ways.

A recursive formula may list the first two (or more) terms as starting values, depending upon the natrue of the sequence. In such caes, the an portion of the formula is dependent upon the previous two (or more) terms..


Always 2 parts!

Recursion is described as
an "iterative" procedure.

Examples:

1.  Write the first four terms of the sequence:   

 

In recursive formulas, each term is used to produce the next term.  Follow the movement of the terms through the set up at the left.

Answer:  -4, 1, 6, 11

2.  Consider the sequence  2, 4, 6, 8, 10, ...
Explicit formula:
Recursive formula:

 

Certain sequences, such as this arithmetic sequence, can be represented in more than one manner.  This sequence can be represented as either an explicit (general) formula or a recursive formula.

 

3Consider the sequence  3, 9, 27, 81, ...
Explicit formula:
Recursive formula:

 

Certain sequences, such as this geometric sequence, can be represented in more than one manner.  This sequence can be represented as either an explicit formula or a recursive formula.

 

4Consider the sequence  2, 5, 26, 677, ...
Recursive formula:

 

This sequence is neither arithmetic nor geometric.  It does, however, have a pattern of development based upon each previous term.

5.  Write the first 5 terms of the sequence 


Notice how the value of n is used as the exponent for the value (-1).  Also, remember that in recursive formulas, each term is used to produce the next term.  Follow the movement of the terms through the set up at the left.

Answer:  3, 15, -75, -375, 1875

 

 

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