Question

Consider the following recurrence relation: H(n) = {0 if n lessthanorequalto 0 1 if n = 1 or n = 2 H(n - 1) + H (n - 2)-H(n - 3) if n > 2. (a) Compute H(n) for n = 1, 2, ...., 10. (b) Using the pattern from part (a), guess what H(100) is. 2. Consider the recurrence relation defined in Example 3.3 (FROM TEXT BOOK, also discussed in class and shown in slides) Suppose that, as in the example, you borrow $500, but you pay her back $100 each week. Each week, Ursula charges you 10% interest on the amount you still owe, after your $100 payment is taken into account (a) Write down a recurrence relation for M(n), the amount owed after n weeks. (b) How much will you owe after four weeks? Let H(n) be as in Example 3.4, and let T(n) be as in Exercise 1.3. Write H(n) in terms of T(n - 1). Explain your reasoning. What is closed form solution; please write and example of closed form solution and its recurrence relation. What is proof by induction and what are the steps to be followed to prove by induction and show an example?

Need answers for 1-5

Answer 1

(a)
H(1) = 1
H(2) = 1
H(3) = 2
H(4) = 2
H(5) = 3
H(6) = 3
H(7) = 4
H(8) = 4
H(9) = 5
H(10) = 5

(b) H(100) will be 100/2 which is 50.