High School Mathematics Extensions/Counting and Generating Functions/Solutions

Counting and Generating FunctionsEdit

At the moment, the main focus is on authoring the main content of each chapter. Therefore this exercise solutions section may be out of date and appear disorganised.

If you have a question please leave a comment in the "discussion section" or contact the author or any of the major contributors.


These solutions were not written by the author of the rest of the book. They are simply the answers I thought were correct while doing the exercises. I hope these answers are useful for someone and that people will correct my work if I made some mistakes.

Generating functions exercisesEdit

1.

(a) 
 
 
 
(b) 
 
 
 
(c) 
 
 
 
(d) 
 
 
 
 
 

2.

(a) 
 
 
 
(b) 
 
 
 
 

2c only contains the exercise and not the answer for the moment

(c) 

Linear Recurrence Relations exercisesEdit

This section only contains the incomplete answers because I couldn't figure out where to go from here.

1.

 

Let G(z) be the generating function of the sequence described above.

 
 
 
 
 
 
 
 

2.

 

Let G(z) be the generating function of the sequence described above.

 
 
 
 
 

3. Let G(z) be the generating function of the sequence described above.

 
 
 
 
 
We want to factorize   into   , by the converse of factor theorem, if (z - p) is a factor of f(z), f(p)=0.
Hence α and β are the roots of the quadratic equation  
Using the quadratic formula to find the roots:
 
In fact, these two numbers are the famous golden ratio and to make things simple, we use the greek symbols for golden ratio from now on.
Note:  is denoted   and   is denoted  
 
By the method of partial fraction:
 
 
 
 
 

Further Counting exercisesEdit

1. We know that

 

therefore

 
Thus
 

2.  

 
Thus
 

*Differentiate from first principle* exercisesEdit

1.