Proofs Self-Assessment Instructions Instructions: These problems are provided for your own self-assessment. You need to work on them alone, solve them without looking at the solution, and only then compare your solution with the provided one. This will be the best way for you to check that you are able to solve these prob- lems at the exam. You do not need to solve all of these problems. Please solve at least 3 of them without mistake. Please make sure that you identify the method of proof that you are using. Submit your evidence of self-assessment by showing your work and self-correction. There are a limited number of exercises. If you need practice, avoid using them, or you might run out of self-assessment problems. Problems Exercise 1. Show that -9n2 + 2n + 6 is odd if n is odd. Exercise 2. Show that 7n' + 5n - 1 is odd for any n e N. Exercise 3. Show that n is odd if 10n2 - 5n -6 is odd. Exercise 4. Show that n is even if and only if 2n2 + 7n + 8 is even. Exercise 5. Show that 8 - n - 3 is even if n is odd. Exercise 6. Show that 7n2 - 3n - 8 is even for any n E N. Exercise 7. Show that n is even if 10n- + 7n - 6 is even. Exercise 8. Show that 7n' + 9n - 9 is odd for any n e N. Exercise 9. Show that -3n2 - 3n + 3 is odd for any n E N. Exercise 10. Show that 7n2 - 2n - 4 is odd if n is odd. Exercise 11. Show that Sn + 8n + 9 is odd for any n e IN. Exercise 12. Show that 5n2 - 4n - 4 is even if n is even. Exercise 13. Show that 2n2 - 5n - 2 is even if n is even. Exercise 14. Show that -5n' + n - 1 is odd for any n e IN. Exercise 15. Show that -5n - 3n - 2 is even for any n e IN. Exercise 16. Show that -7n2 + 3n -5 is odd for any n e INN. Exercise 17. Show that n is even if -3n2 + 2n - 9 is odd. Exercise 18. Show that -5n + 9n + 7 is odd for any n e IN. Exercise 19. Show that -5n12 - n + 2 is even for any n e IN. Exercise 20. Show that n2 - 5n - 10 is even for any n E IN. 1