Final answer:
To solve the given recurrence relation using generating functions, one must find both the homogeneous and particular solutions, typically involving the use of series expansions such as the binomial theorem and manipulation of series. The initial conditions are required for a complete solution.
Step-by-step explanation:
The recurrence relation given is an - 10an-1 + 21an-2 = 3. To solve this using generating functions, we would typically first find the homogeneous solution, which solves the case where the right side is zero, and then find the particular solution for the non-homogeneous part (the constant 3, in this case). Unfortunately, this question does not provide initial conditions, which are required to find the complete solution.
To solve the homogeneous part, we would assume a generating function G(x) such that its coefficients are the terms of the sequence, leading to an algebraic equation involving G(x). Solving this equation would give us G(x), and we could then use series expansion to obtain the closed form of an. The non-homogeneous solution would involve special techniques for dealing with the constant term, such as looking for a constant solution.
The details of solving such equations involve significant algebra and manipulation using series expansions like the binomial theorem and methods for series manipulations.