Due at 4:59pm in my mailbox in SAS 3151, Thursday, February 2, 2017

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 rational function F_n(x) in the variable x,
defined by the quadratic recurrence F_0 = x+(1/x), F_{n+1}(x) = F_n(x)^2 - 2 for all n ≥ 0 one has

F_n(x) = x^(2^n) + x^(-2^n).

- 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 20(e), page 25.
- Please compute integers x,y,z such that 35x + 45y + 63z = 1.
- 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*