1. 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

2. 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) : 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

3. F k denotes the kth term of the Fibonacci sequence (F 1 = 1, F 2 = 1 and F k = F k 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