Consider the recurrence xk+2 = axk+1 + bxk + c(k) (*) where c(k) is a function of

Question:

Consider the recurrence
xk+2 = axk+1 + bxk + c(k) (*)
where c(k) is a function of k, and consider the related recurrence
xk+2 = axk+1 + bxk (**)
Suppose that xk = pk is a particular solution of (*).
(a) If qk is any solution of (**), show that qk + pk is a solution of (*).
(b) Show that every solution of (*) arises as in (a) as the sum of a solution of (**) plus the particular solution pk of (*).