Use the Principle of Mathematical Induction in the following Exercises. 17. A sequence al, a2, аз numbers k22. Show that a3.7H for all natural numbers. is defined by letting al-3 and a,-7aH for all natural 18. A sequence bo b,b, is defined by letting bo- 5 and b.4+b.1 for all natural numbers k. Show that b54n for all natural number n using mathematical induction 19. A sequence d, d,, d, ...is defined by letting d 2 and d-for all natural numbers, k22. Show that d- for all n E N 7n 20. Prove that for all n E N cos α+cos (α + β) + cos (α + 2B) + cos (α + (n-1) β) sin sin 2 e 2" sin θ sín n θ . 21. Prove that, cos θ cos 2θ cos29 cos2-e , for a all n e N (n+1) sin 22, Prove that, sinθ+sin 2θ+sin 3θ + + sinn®- for alln E N sin 23. Show thatis a natural number for all neE N. 5 3 15 1 13 n+1 n+2 2n 24 24. Prove that, for all natural numbers >1 25. Prove that number of subsets of a set containing n distinct elements is 2, for all

Use generalized induction and Exercise 43 to prove that n22n for all integers n5. (In connection with this result, see the discussion of counterexamples in the Appendix.) 1+2n2n for all integers n3

Given the recursively defined sequence a1=1,a2=4, and an=2an1an2+2, use complete induction to prove that an=n2 for all positive integers n.

Prove by induction that n2n.