Use mathematical induction to prove that the formula is true for all natural numbers m. {image} Choose the first step of the proof from the following: 1. 2. (for m = 1) {image} it is true let for m = p it is true {image} Prove for m = p + 1 : {image} None of these choices 3. 4. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Show that n 2 - n + 73 is odd for all natural numbers n Show that n 2 - n + 73 is odd for all natural numbers n. Choose the second step of the proof from the following: (n = 1) : 1 - 1 + 73 is odd let for n = p : p 2 - p + 73 is odd prove for n=p+1: (p + 1) 2 - (p - 1) + 73 is odd None of these choices 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

F k denotes the kth term of the Fibonacci sequence (F 1 = 1, F 2 = 1 and F k = F k - 1 + F k - 2). Let a k be the kth term of the sequence defined recursively by {image} Find a formula for b k in terms of the Fibonacci numbers F k. Choose the correct answer from the following: 1. {image} None of these choices 2. 3. 4. 5. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50