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Prove each statement in 8-23 by mathematical induction. 8. 5"-1 is divisible by 4, for every integer n ≥ 0. 9. 7-1 is divisible by 6, for each integer n ≥ 0. 10. n³-7n+3 is divisible by 3, for each integer n ≥ 0.

Prove each statement in 8-23 by mathematical induction. 8. 5"-1 is divisible by 4, for every integer n ≥ 0. 9. 7-1 is divisible by 6, for each integer n ≥ 0. 10. n³-7n+3 is divisible by 3, for each integer n ≥ 0.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 42E
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Help me in understanding #10 please. Explain as thuroghly as possible.

298 CHAPTE by buying a collection of 5-stamp packages and 8-stamp packages. 4. For each positive integer n, let P(n) be the sentence that describes the following divisibility property: 5"-1 is divisible by 4. a. Write P(0). Is P(0) true? b. Write P(k). c. Write P(k+1). d. In a proof by mathematical induction that this di- visibility property holds for every integer n ≥ 0, what must be shown in the inductive step? 5. For each positive integer n, let P(n) be the inequality 2" < (n+1)!. a. Write P(2). Is P(2) true? b. Write P(k). c. Write P(k+ 1). d. In a proof by mathematical induction that this inequality holds for every integer n ≥ 2, what must be shown in the inductive step? 6. For each positive integer n, let P(n) be the sentence Any checkerboard with dimensions 2 X 3n can be completely covered with L-shaped trominoes. a. Write P(1). Is P(1) true? b. Write P(k). c. Write P(k+ 1). d. In a proof by mathematical induction that P(n) is true for each integer n ≥ 1, what must be shown in the inductive step? 7. For each positive integer n, let P(n) be the sentence In any round-robin tournament involving n teams, the teams can be labeled T₁, T2, T3,..., Tn so that T; beats Ti+1 for every i = 1, 2,..., n-1. a. Write P(2). Is P(2) true? b. Write P(k). c. Write P(k+1). d. In a proof by mathematical induction that P(n) is true for each integer n ≥ 2, what must be shown in the inductive step? Prove each statement in 8-23 by mathematical induction. 8. 5"-1 is divisible by 4, for every integer n ≥ 0. 9. 7-1 is divisible by 6, for each integer n ≥ 0. 10. n³-7n+3 is divisible by 3, for each integer n ≥ 0. 11. 32n 12. For H 13. For who H 14. n³ 15. n(n 16. 2" 17. 1- 18. 5" 2 19. n 20. 22 21. V o n 22. 1 ar 23. a. b 24. A a S 25. F b 26. 27. 28. S Exer leng for 29.
Transcribed Image Text:298 CHAPTE by buying a collection of 5-stamp packages and 8-stamp packages. 4. For each positive integer n, let P(n) be the sentence that describes the following divisibility property: 5"-1 is divisible by 4. a. Write P(0). Is P(0) true? b. Write P(k). c. Write P(k+1). d. In a proof by mathematical induction that this di- visibility property holds for every integer n ≥ 0, what must be shown in the inductive step? 5. For each positive integer n, let P(n) be the inequality 2" < (n+1)!. a. Write P(2). Is P(2) true? b. Write P(k). c. Write P(k+ 1). d. In a proof by mathematical induction that this inequality holds for every integer n ≥ 2, what must be shown in the inductive step? 6. For each positive integer n, let P(n) be the sentence Any checkerboard with dimensions 2 X 3n can be completely covered with L-shaped trominoes. a. Write P(1). Is P(1) true? b. Write P(k). c. Write P(k+ 1). d. In a proof by mathematical induction that P(n) is true for each integer n ≥ 1, what must be shown in the inductive step? 7. For each positive integer n, let P(n) be the sentence In any round-robin tournament involving n teams, the teams can be labeled T₁, T2, T3,..., Tn so that T; beats Ti+1 for every i = 1, 2,..., n-1. a. Write P(2). Is P(2) true? b. Write P(k). c. Write P(k+1). d. In a proof by mathematical induction that P(n) is true for each integer n ≥ 2, what must be shown in the inductive step? Prove each statement in 8-23 by mathematical induction. 8. 5"-1 is divisible by 4, for every integer n ≥ 0. 9. 7-1 is divisible by 6, for each integer n ≥ 0. 10. n³-7n+3 is divisible by 3, for each integer n ≥ 0. 11. 32n 12. For H 13. For who H 14. n³ 15. n(n 16. 2" 17. 1- 18. 5" 2 19. n 20. 22 21. V o n 22. 1 ar 23. a. b 24. A a S 25. F b 26. 27. 28. S Exer leng for 29.
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