Worked Problems 11-14 from Appendix 3

These problems have 3 parts. Give the generating function, SUM(a_k x^k : k=0...), a name, some g(x) and using the recurrence relation and initial conditions express g(x) as some function like (using 2 and 5 for example) [c+dx]/(1-2x)(1-5x) .

Next we use some simple algebra and solving 2 equations in two unknowns ideas to express this function in the form A/(1-sx)

B/(1-tx) = [A(1-tx) + B(1-sx)] / (1-sx)(1-tx) which we set equal to [c+dx]/(1-2x)(1-5x) and then, using just the numerators, we solve for A,B.

Finally we realize that 1/(1-sx) is just SUM( 2^k x^k : k=0...) , hence
g(x) = SUM( (A*2^k + B5^k) x^k : k=0...), which just means that a_k = A*2^k + B*5^k for every k.