Problem 1. [Sundstrom # 4.1.3(a-b)] Use mathematical induction to prove each of the following: n(3n+ 1) 2 (b) For each natural number n, 1+5+9+---+ (4n3) =n(2n 1) (a) For each natural number n, 2+5+8+---4+ (3n1) = Problem 2. [Sundstrom #4.1.7(a-f)] Back in homework 4 (and 3.1 in the textbook), we defined congruence modulo n for a natural number n. In Section 3.5, the following rather intuitive result is proven as Corollary 3.32: Corollary 3.32: If n N, then each integer is congruent, modulo n, to precisely one of the integers 0,1,2,...,n 1. That is, for each integer a, there exists a unique integer r such that a=r (modn)and 0 = r (mod 3) and r {0,1,2}. (c) Find the value of r so that 4> = r (mod 3) and 7 {0, 1,2}. (d) For two other values of n, find the value of r so that 4 = r (mod 3) and r {0, 1, 2}. (e) If n N, make a conjecture concerning the value of r where 4" = r (mod 3) and r {0,1,2}. This conjecture should be written as a self-contained proposition including an appropriate quantifier. (f) Use mathematical induction to prove your conjecture. Problem 3. [Sundstrom #4.1.8(b)] Use mathematical induction to prove that: 3 For each natural number n, 6 divides (n n) Problem 4. [Sundstrom #4.2.1(a)] Use mathematical induction to prove that: For each natural number n with n > 2,3" > 142&quot