COS 550 F08 HW #1 DUE Thursday 9/11/08
Problem 1 Below, 20 points
Problem 2 Below, 20 points
Problem 3 Below, 20 points
Problem 4 Below, 20 points
Problem 5 Below, 20 points
********** Problem Notes *********,

Below ^ means exponentiation. Thus 2^3 is 2 to the third power which is 
equal to 8.

PROBLEM 1: Complete the proof by induction that was started in class.

PROBLEM 2: Prove for any natural number 11^(n+2) + 12^(2n+1) is divisible 
by 133.

PROBLEM 3: Prove that 1^2 + 2^2 + 3^2 +...+ n^2 = n(n+1)(2n+1)/6.

PROBLEM 4: Prove that the sum of 3 consecutive cubes is divisible by 9.

PROBLEM 5: Find some formula that gives the number of ways to parenthesize 
a product of n factors. For example, if n = 2, you can only write (a*b). 
For n =3 you have (a*b)*c and a*(b*c). For n = 4 you have a*(b*(c*d)), 
a*((b*c)*d), (a*b)*(c*d), (a*(b*c))*d, ((a*b)*c)*d. If you can't come up 
with a neat formula, produce an efficient program that can compute this 
number.