1. (3 marks) Find a formula for the nth term of the sequence defined by the third-

order recursion

tn+3  = 7tn+1  + 6tn

with initial values  t0  = 2, t1  = − 8, t2  = 0.  Calculate t3 directly and use it to

check your formula.

3. (3 marks) I want to construct a train of total length 20. I have three kinds

of cars, A and B cars each have length 2 and C cars have length 3.  For

example, here is one possibility using 3 A-cars, 1 B-car and 4 C-cars,.

Other possibilities are obtained by rearranging the cars, or using different

numbers of cars of each type. Note that a train has a front and a back, so that

the mirror image of the train diagrammed above is a different train.

How many different ways are there to make such a train?

[You can solve this any way you wish but you need to explain how you solved it.]

4. (3 marks)

(a) Calculate A2023v where A = [ 3     3] and v = [2(0)].

[Hint: if v were an eigenvector of A, you could do it.  Use the linear combination idea of Ex 414 and 415.]

(b)  Let A2023  = [c(a)    d(b)].  Calculate b.

To get full marks your answers must be in the simplest possible terms.

5. (3 marks) I want to pave a 2×n rectangle with 1×2 blocks which come

in two colours, white and grey.  Let wn be the number of different ways

this can be done.  For example, it can be shown that w8 = 8704 and the

diagram at the right displays one of these pavings.

(a)  Find a linear recursive equation of order 2 for wn  and provide the values of w1  and w2  as initial   conditions. Use the recursion to calculate w3 and then check your result by direct enumeration of the pavings of a 2×3 rectangle. [Note: w3 is large enough that you do not want to draw all of the pavings. But you can classify them into different types and enumerate the pavings of each type easily enough.

(b) Now take the same wall-building problem but suppose all the tiles are the same colour. Find a formula for xn , the number of different ways this can be done. [You do not need to show your work as we have     already done the analysis in class.]

(c) Is there a simple relationship between xn  and wn? Does your solution for xn  in part (b) allow you to write down a formula for wn? As a hint, compare the values of x8  and w8 .