13.10. suppose that a sequence (ao, a1, a2, ) of real numbers satisfies the recurrence relation an -5an-1+6an-20 for all n> 2. (a) what is the order of the linear recurrence relation? (b) express the generating function of the sequence as a rational function. (c) find a generic closed form solution for this recurrence relation. (d) find the terms ao, . ,a5 of this sequence when the initial conditions are given by ao 2 and a5 (e) find the closed form solution when ao 2 and a 5.

a. This recurrence is of order 2.

b. We're looking for a function such that

Take the recurrence,

Multiply both sides by and sum over all integers :

Pull out powers of so that each summand takes the form :

Now shift the indices and add/subtract terms as needed to get everything in terms of :

Solve for :

c. Splitting into partial fractions gives

Recall that for , we have

so that for and , or simply , we have

which means the solution to the recurrence is

d. I guess you mean and , in which case

e. We already know the general solution in terms of and , so just plug them in:

divide by 5

step-by-step explanation:

60/5 = 12

6, 10

9, -13

12, -21

18, -23

step-by-step explanation:

you just add 6 to the x-coordinate and subtract 10 from the y-coordinate.