What is the general form of the particular solution, guaranteed to exist by Theorem 6, of the linear non- homogeneous relation an = r(n) + F(n) where r(n) is some homogeneous recurrence relation with roots 1, 3, and 3, and where F(n) = n² . 3" + 3"+2?

Determine the multiplicity of the non-homogeneouus term for the following recurrence relation f(n) = 2f(n-1) - f(n-2) +1

1. Use method of undetermined coefficients to find the solution of non-homogeneous Fi- bonacci recurrence relation [ 4 marks ] F, = Fn-1+ Fn-2+ 2", n 2 2 with initial conditions Fo = 0 and F, = 1

Let an be the number of strings (i.e. sequences of letters, non-sense words) of length n that can be formed using the four letters Q,X,T and Z. (a)Find an explicit formula for an (b)Find a recurrence relation that is satisfied by the numbers an. (c) Find a rational generating function (ratio of two polynomials) that is a generating function for the numbers an.