CONCEPTS 9. 1' + 2 + 3 3 + . . . + 1) = n (n + 1)2 1. Mathematical induction is a method of proving that a statement 4 P(n) is true for all numbers n. In Step I we prove 10. 13 + 3 + 53 + . .. + (2n - 1)3 = n'(2n2 - 1) that is true. 11. 25 + 43 + 63 + . .. + (2n)3 = 2n*(n + 1)- 2. Which of the following is true about Step 2 in a proof by 1 12. 1 + ...+ mathematical induction? 1 .2 2-3 3-4 n(n + 1) (n + 1 ) (i) We prove "P(k + 1) is true." 13. 1 - 2 + 2 .23 + 3 - 23 + 4 - 24 + . . . + n. 2" (ii) We prove "If P(k) is true, then P(k + 1) is true." = 2[1 + (n - 1)2"] 14. 1 + 2 + 22 + . . . + 20-1 = 2" - 1 SKILLS 15. Show that n + n is divisible by 2 for all natural numbers n. 3-14 - Use mathematical induction to prove that the formula is 16. Show that 5" - 1 is divisible by 4 for all natural numbers n. true for all natural numbers n. 17. Show that n - n + 41 is odd for all natural numbers n. +. 3. 2 + 4 + 6 + . . . + 2n = n(n + 1) 18. Show that n- n + 3 is divisible by 3 for all natural n(3n - 1) numbers n. 4. 1 + 4 + 7 + . . . + (3n - 2) = 2 19. Show that 8" - 3" is divisible by 5 for all natural numbers n. n(3n + 7) 20. Show that 32" - 1 is divisible by 8 for all natural numbers n. 5. 5 + 8 + 11 + . . . + (3n + 2) = 2 . 21. Prove that " -1, then (1 + x)" 2 1 + nx for all natural 7. 1 . 2 + 2 . 3 + 3 . 4 + . . . + n(n + 1 )= n(n + 1 )(n + 2) numbers n. 3 24. Show that 100n = n for all n 2 100. n(n + 1)(2n + 7) 8. 1 .3 + 2 . 4 + 3 - 5 + . . . + n(n + 2)= 25. Let a,+1 = 3a, and a, = 5. Show that a,, = 5 . 3" for all nat- 6 ural numbers n