Recursive definitions

(See attached file for full problem description) --- Give a recursive definition of a) the sequence {an}, n=1,2,3,…if i. an = 1+(-1)n ii. an = n2 b) of the set of ordered pairs of positive integers S = {(a,b) | a є Z+, b є Z+, and 3 |(a+b)}.

Counting: How many positive integers less than 1000?

How many positive integers less than 1000? a) have distinct digits b) have distinct digits and are even c) divisible by 7 d) divisible by 7 and not 11 c) both 7 and 11 d) either 7 or 11 e) exactly one of 7 or 11 f) neither 7 or 11

Recursive definitions

(See attached file for full problem description) --- Give a recursive definition of a) of the functions max and min so that mx{a1,a2,..an and min {a1,a2,…an} are the maximum and minimum of the n numbers a1,a2,…an respectively b) prove that f12+f22+..fn2 = fnfn+1 whenever n is a positive integer fn is the Fibonacci sequence.

Counting

1. how many license plates can be made using three letters followed by the three digits of four letters followed by two digits 2. how many bit strings of length 10 contain either 5 consecutive 0s or five consecutive 1s.

Show that if 7 integers are selected from the first 10 positive integers there must be at least 2 pairs of these integers with the sum 11. Is the conclusion true if 6 integers are selected instead of 7 How many numbers must be selected from the set {1,3,5,7,9,11,13,15} to guarantee that at least one pair of these numbers add up to 16.

Show that if 7 integers are selected from the first 10 positive integers there must be at least 2 pairs of these integers with the sum 11 Is the conclusion true if 6 integers are selected instead of 7 How many numbers must be selected from the set {1,3,5,7,9,11,13,15} to guarantee that at least one pair of these numbers add up to 16.