1. Let ar = 3* + k - 2 for all k > 0. a. Write down the values of @1, 02 and a3. b. Write down the values of A (1), A(2) and A(3) defined by the recurrence relation: A(0) = -1, A(k) = 3A(k - 1) - 2k + 7, k 21 c. Show that A( k ) = ax is a solution of the recurrence relation for all values of k > 1. 2. Write down all derangements of the set { a, b, c, d} and show that the number of derangements is the same as predicted by the recurrence D(n) = (n - 1)(D(n - 2) + D(n - 1)) with initial values D(1) = 0 and D(2) = 1. Hint: a derangement is a permutation of an ordered set where no element is in the same place as before. Example: {b, a, d, c} is a derangement of {a, b, c, d'} because all of the letters positions have changed. 3. Solve the recurrence relation A(n) = 6A(n - 1) - 11A(n - 2) + 6A(n - 3) subject to initial values A(1) = 2, A(2) = 6, A(3) = 20. Hint: See example 2.4.6 on page 141 of the course text