1. Prove that 2 - 2 . 7 + 2 .72 -. . .+2(-7)" - (1 -(-7) "#1)/4 whenever n is a nonnegative integer. 2. Prove that 1 . 1! + 2 . 2!+. . .+n . n! =(n + 1)! -1 whenever n is a positive integer. 3. How many bit strings are there of length six or less, not counting the empty string? 4. How many strings are there of four lowercase letters that have the letter x in them? 5. Find f (1), f (2), f (3), f (4), and f (5) if f (n) is defined recursively by f (0) = 3 and for n = 0, 1, 2, . .. a) f (n + 1) = -2f (n). b) f (n + 1) = 3f (n) + 7. c) f (n + 1) =f (n)2 - 2f (n) -2. d) f (n + 1) = 3f(n)/3 6. Give a recursive definition of the sequence {an} , n = 1, 2, 3, .. . if a) an = 4n - 2. b) an = 1 + (-1)" c) an = n(n + 1). d) an = n. 7.For the set {u, v, w, x}, how many permutations are there ? How many permutations start with u? Please write down all permutations that start with u? 8. A bowl contains 10 red balls and 10 blue balls. A person selects balls at random without looking at them. a) How many balls must she select to be sure of having at least three balls of the same color? b) How many balls must she select to be sure of having at least three blue balls