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 a_{1.}
2. the recursion equation for a_{n} as a function of
a_{n}_{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 a_{n} 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.


3.
Consider 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.


4.
Consider 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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sequences and series.
Click here. 

