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INTRODUCTION TO RECURSION AND SEQUENCES
Jump right to the exercises!
DEFINITION: sequence; notation for sequences
A sequence is an ordered list of numbers.
Each number in the sequence is called a term.
The nth term can be denoted as either
un
(subscript notation)
or
u(n)
(function notation).
|
EXAMPLE:
The first five terms of the sequence defined by
u(n)=
n2
are:
u(1)=
12
=1
u(2)=
22
=4
u(3)=
32
=9
u(4)=
42
=16
u(5)=
52
=25
EXAMPLE:
The 27th term of the sequence
defined by
un
=n+3
is:
u27
=27+3=30
DEFINITION: recursion
Recursion is a process in which each step of a pattern is dependent on the step or steps
that came before it.
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DEFINITION: recursive formula
A recursive formula must specify:
- one (or more) starting terms
- a recursive rule that defines the nth term in relation to previous term(s)
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EXAMPLES:
The recursive rule
u1
=2 ;
un
=u
n-1
+3 for
n≥2
generates the sequence
2, 5, 8, 11, 14, …
Thought process: Start with the number 2. To find any other term, take the previous term and add 3.
The recursive rule
w1
=1,
w2
=1
wn
=w
n-1
+
w
n-2
for
n≥3
generates the sequence
1, 1, 2, 3, 5, 8, 13, …
Thought process: Start with the numbers 1 and 1. To find any other term, take the previous two terms
and add them together.
Some sequences can be defined both recursively and non-recursively.
For example, the sequence
3, 5, 7, 9, 11, …
can be defined in either of the following ways:
as a recursive sequence:
u1
=3 ;
un
=u
n-1
+2 for
n≥2
as a nonrecursive sequence:
un=2n
+
1 for
n≥1
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.