Need help with problems 48 and 56, (discrete structures) chapter 2 section 2

n 1 + 1 2 + 46. Prove that 1 forn21 3 (Note that the denominators increase by 1, not by powers of 2.) For Exercises 47-58, prove that the statements are true for every positive integer. 47. 23" - 1 is divisible by 7. 48. 32 + 7 is divisible by 8. 49.7" - 2" is divisible by 5. 50.13" - 6" is divisible by 7. 51.2" + (-1)"+' is divisible by 3. 52.25 + 1 + 5*+2 is divisible by 27. 53.341 +2 +524+ is divisible by 14. 54.721 + 16n - 1 is divisible by 64 55. 10" + 3.4"*2 + 5 is divisible by 9. 56. - is divisible by 3. 57.7 + 2n is divisible by 3. 58. X" - 1 is divisible by x - 1 for x + 1. 126 Proofs, Induction, and Numbe 59. Prove DeMoivre's Theorem: (cose + i sin o)" = cos no + i sin ne for all n 2 1. Hint: Recall the addition formulas from trigonometry: cos(a + B) = cos a cosB - sin a sin B sin(a + b) = sin a cosB + cos a sin B 60. Prove that sin ne sin e + sin 30 +++++ sin(2n-1) for all n 2 1 and all e for which sin 0 + 0. sin 8 n 1 + 1 2 + 46. Prove that 1 forn21 3 (Note that the denominators increase by 1, not by powers of 2.) For Exercises 47-58, prove that the statements are true for every positive integer. 47. 23" - 1 is divisible by 7. 48. 32 + 7 is divisible by 8. 49.7" - 2" is divisible by 5. 50.13" - 6" is divisible by 7. 51.2" + (-1)"+' is divisible by 3. 52.25 + 1 + 5*+2 is divisible by 27. 53.341 +2 +524+ is divisible by 14. 54.721 + 16n - 1 is divisible by 64 55. 10" + 3.4"*2 + 5 is divisible by 9. 56. - is divisible by 3. 57.7 + 2n is divisible by 3. 58. X" - 1 is divisible by x - 1 for x + 1. 126 Proofs, Induction, and Numbe 59. Prove DeMoivre's Theorem: (cose + i sin o)" = cos no + i sin ne for all n 2 1. Hint: Recall the addition formulas from trigonometry: cos(a + B) = cos a cosB - sin a sin B sin(a + b) = sin a cosB + cos a sin B 60. Prove that sin ne sin e + sin 30 +++++ sin(2n-1) for all n 2 1 and all e for which sin 0 + 0. sin 8