I need help with those question

Mathematics 311 Exam 2 Practice October 18, 2019 Name: 1. Prove that there do not exist integers a, b, and c satisfying a2 + b2 = 4c + 3. 2. Prove that 2" g (n + 1)! for all n. 3. (a) Prove that the product of three consecutive natural numbers is divisible by 6. (b) Prove that the product of four consecutive natural numbers is divisible by 24. 4. Recall that 5!." = {0,1,2,3,...}. Prove that if n E N and n 2 18, then there exist 3;, y E Z+ satisfying n = 4s; + 73;. 1 5. Dene a sequence {on} recursively by :11 = 1 and on\" = 3 . an (a) Write out the rst six terms of the sequence. (b) Prove that 1 S on