2. Recall that the Fibonacci sequence F(n) is defined by the recurrence relation F(0) = 0, F(1) = 1, and F(n) = F(n 1) + F(n 2) for n > 1.

a. Show that if a = (1+ 5)/2 and b = (1 5)/2 then a 2a1 = b 2b1 = 0. Conclude that a 2 = a + 1 and that b 2 = b + 1.

b. Prove via induction that F(n) = a n b n a b .

3. Let L be the language {Austin, Houston, Dallas}. How many elements are there in L ?

4. Let = {a,b,c}. How many elements are there in ?

5. Let L = {w {a,b} : every a in w is immediately followed by a b}. List the first six elements in a lexigraphic enumeration of L.

6. Let L = {w {1,2,3} : |w| is even}. List the first twelve elements in a lexigraphic enumeration of L.

1. Prove via induction that n k" = n(n + 1)(2n + 1) a. fi n+1 )=T. k2 k-2