High School Mathematics Extensions/Counting and Generating Functions/Solutions

Counting and Generating Functions 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 exercises 1. (a) (b) (c) (d) 2. (a) (b) 2c only contains the exercise and not the answer for the moment (c) Linear Recurrence Relations exercises 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 exercises 1. We know that therefore Thus 2. Thus *Differentiate from first principle* exercises 1.