Introduction to Recursion and Sequences

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Algebra II Table of Contents

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INTRODUCTION TO RECURSION AND SEQUENCES

DEFINITION: sequence; notation for sequences
A sequence is an ordered list of numbers.
Each number in the sequence is called a term.
The n^th 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 27^th 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.

DEFINITION: recursive formula
A recursive formula must specify:

one (or more) starting terms
a recursive rule that defines the n^th term in relation to previous term(s)

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.

Algebra II Table of Contents

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