CS 4521 Fall Semester, 2009

15 Points

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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!) = Ω(n lg n), which follows from lg(n!) ≥ 1/4 n lg(n) (for n ≥ 4).

Then lg(n!) = Θ(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(2^{n}) and n! = o(n^{n}), you can show that the limits as n goes to infinity of n!/2^{n}and n!/n^{n}are infinity and 0 respectively, since n!, 2^{n}, and n^{n}are all (asymptotically) nonnegative. - (3 points) Exercise 3.2-7, page 57.
Hint: use the fact that φ satisfies the equation

x^{2}= x + 1 -- you need to prove this (and use a "strong" form of induction). - (3 points) Exercise 4.3-1, page 75.
- (2 points) Exercise 4.3-3, page 75.