11. For all integers n, prove the following:

(a) 2n³ + 3n² + n is an integral multiple of 6.

(b) n⁵ - 5n³ + 4n is an integral multiple of 120.

12. Prove that n(n² - 1)(3n + 2) is an integral multiple of 24 for all integers n.

13. Guess a formula for each of the following and prove it by mathematical induction:

(a)

(b)

14. Guess a formula for each of the following and prove it by mathematical induction:

(a)

(b)

15. Guess a simple expression for the following and prove it by mathematical induction:

16. Find a simple expression for the product in Problem 15, using the factorization

17. Prove the following properties of the Fibonacci numbers F_n for all integers n greater than or equal to 0:

(a)

(b)

(c)
 

Saturday, April 11, 1998