Fibonacci Numbers

(1) Definition (Fibonacci Numbers) The Fibonacci sequence is defined recursively by and for The terms of this sequence ar called the Fibonacci numbers.

(2) Example (Fibonacci Numbers) Write out the first 21 Fibonacci numbers.

    Solution. The first 21 Fibonacci numbers are

The first 1000 Fibonacci numbers are listed on the web page titled Fibonacci Sequence .

(3) Example (Fibonacci Numbers) For all positive integers
    
    Solution. For so the base case holds. For some integer assume we will show We have,
    








Therefore, by mathematical induction, for all positive integers     

(4) Example (Fibonacci Numbers) For all positive integers
    
    Solution. For so the base case holds. For some integer assume we will show This is accomplished by,
    


Therefore, by mathematical induction, for all positive integers     

(5) Example (Fibonacci Numbers) For all positive integers
    
    Solution.  For so the base case holds. For some integer assume we will show This is accomplished by,
    

    
Therefore, by mathematical induction, for all positive integers     

(6) Example (Fibonacci Numbers) For all positive integers
    
    Solution.  For so the base case holds. For some integer assume we will show This is accomplished by,
    

    
Therefore, by mathematical induction, for all positive integers     

(7) Example (Fibonacci Numbers) For all positive integers
    
    Solution. For so the base case holds. For some integer assume we will show This is accomplished by,











Therefore, by mathematical induction, for all positive integers     

(8) Example (Fibonacci Numbers) For all positive integers

   
    
    Solution. For so the base case holds. For some integer assume We will show



Using the induction hyposthesis,    
    













    
Therefore, by mathematical induction, for all positive integers     

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Cite this as:
Fibonacci Numbers
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/fibonacci-numbers.html