The assignment:
Do the following Exercises from the text, which ask you to prove properties of n!, the Fibonacci numbers, and polynomials.

• (7 points) Exercise 3.2-3, page 57. First show that lg(n!) = O(n lg n), which follows from lg(n!) ≤ n lg(n) (easy to show).
  Second show lg(n!) = Omega(n lg n), which follows from lg(n!) ≥ 1/4 n lg(n) (for n ≥ 4).
  Then lg(n!) = Theta(n lg n) follows from "First" and "Second" and Theorem 3.1, page 46).
  Hints for "Second": first show that lg(n!) ≥ ceiling((n+1)/2) lg(ceiling(n/2)) which is ≥
  (n/2) lg(n/2) (show this for both odd and even n), then show that (n/2) lg(n/2) ≥ 1/4 n lg(n) for n ≥ 4.
  To show n! = ω(2ⁿ) and n! = o(nⁿ), you can show that the limits as n goes to infinity of n!/2ⁿ and n!/nⁿ are infinity and 0 respectively, since n!, 2ⁿ, and nⁿ are all (asymptotically) nonnegative.
• (3 points) Exercise 3.2-7, page 57. Hint: use the fact that φ satisfies the equation x² = x + 1 -- you need to prove this (and use a "strong" form of induction).
• (5 points) Problem 3-1, parts a., b., and c., pages 57-58. Hint: for part c., use parts a., b., and Theorem 3.1.