Due at 4:59pm in my mailbox in SAS 3151, Monday, February 8, 2010

All solutions must be submitted in hardcopy
either to me in class or placed in my mailbox.

Note my office hours on my
schedule.

- Prove by induction that for the Fibonacci number
f_n, f_0 = f_1 = 1, f_{n+2} = f_{n+1} + f_n for all n ≥ 0 one has

f_n ≥ ((1 + sqrt(5))/2)^n / sqrt(5) for all n ≥ 0.

- Consider
*C*from ENT, §1.2, Problem 10, page 12. Prove that_{n}*C*for all_{n}= binomial(2n,n) - binomial(2n,n-1)*n ≥ 1*, thus showing that the*C*are integers._{n} - ENT, §2.2, Problem 15, page 25.
- ENT, §2.3, Problem 2(c), page 31. Please compute the Bezout coefficients by the extended Euclidean algorithm presented in class, not the back-substitution given in the book.
- Prove for the extended Euclidean algorithm described in class:
*s*for all_{k}t_{k+1}- s_{k+1}t_{k}= (-1)^{k+1}*-1 ≤ k ≤ n*