18. Discover and prove formulas similar to those of Problem 17 for the Lucas numbers L_n.

19. Use Example 4, in the text above, to prove the following properties of the Lucas numbers for n = 0, 1, 2, ... , and then prove them for all negative integers n.

(a)

(b)

(c)

(d)

20. State an analogue of Example 4 for the Fibonacci numbers instead of the Lucas numbers and use it to prove analogues of the formulas of Problem 19.

21. In each of the following parts, evaluate the expression for some small values of n, use this data to make a conjecture, and then prove the conjecture true for all integers n.

(a)

(b)

(c) F_n-1 + F_n+1.
 

22. Discover and prove formulas similar to the first two parts of the previous problem for the Lucas numbers.

23. Prove the following for all integers m and n:

(a)

(b)

24. Prove that (F_n+1)² + (F_n)² = F_2n+1 for all integers n.
 

25. Let a and b be the roots of the quadratic equation x² - x - 1 = 0. Prove that:

(a)

(b)

(c)

(d)

26. The sequence 0, 1, ½, 3/4, 5/8, 11/16, ... is defined by

Discover and prove a compact formula for u_n as a function of n.
 

Saturday, April 11, 1998