6.35. Give a proof by m inimum counterexample that 1 + 3 + 5 + • • • + (2n — 1) = n2 for every positive integer n.
6.36. Prove that 5 | (n5 — n ) for every integer n.
6.37. Use proof by minimum counterexample to prove that 3 | (2" + 2"+1) for every nonnegative integer n.
6.38. Give a proof by minimum counterexample that 2" > n2 for every integer n > 5.
6.39. Prove that 12 | (n4 — n2) for every positive integer n.
6.40. Let S = {2'' : r e Z, r > 0}. Use proof by minimum counterexample to prove that for every n e N, there exists a subset S„ o f S such that Y l,rs „ ' — n ■
Section 6.4: The Strong P rin cip le of M athem atical Induction
6.41. A sequence {a„} is defined recursively by a\ = 1 and a„ = 2an- \ for n > 2. Conjecture a formula for a„ and verify that your conjecture is correct.
6.42. A sequence [a,,] is defined recursively by «i = 1, «2 = 2 and an = <7„_i + 2a „_2 for n > 3.
Conjecture a formula for an and verify that your conjecture is correct.
6.43. A sequence {a,,} is defined recursively by a\ = 1, = 4, <23 = 9 and a„ = an- 1 - a „_2 + fln-3 + 2(2« - 3) for n > 4. Conjecture a formula for a„ and prove that your conjecture is correct.
6.44. Consider the sequence F\, F 2, F 3, . . . , where
F i = 1, F 2 = 1, F 3 = 2, F 4 = 3, F 5 = 5 and Fg = 8.
The terms o f this sequence are called F ibonacci n u m b ers.
(a) Define the sequence of Fibonacci numbers by means of a recurrence relation.
(b) Prove that 2 | F„ if and only if 3 | n.
6.45. Use the Strong Principle o f Mathematical Induction to prove that for each integer /? > 12, there are nonnegative integers a and b such that n = 3a + lb .
6.46. Use the Strong Principle of M athem atical Induction to prove the following. Let S = {/' e Z : i > 2} and let P be a subset of S with the properties that 2, 3 e P and if n e S, then either n e P or n = ab, where a, b e S. Then every element of S either belongs to P or can be expressed as a product o f elements o f P.
[Note: You might recognize the set P of primes. This is an im portant theorem in mathematics, which appears as Theorem 11.17 in Chapter 11.]
6.47. Prove that there exists an odd integer m such that every odd integer n with n > m can be expressed either as 3a + 11 b or as 5c + I d for nonnegative integers a, b, c and d.
ADDITIONAL EXERCISES FOR CHAPTER 6
6.48. Prove that 1 ■2 + 2 - 3 + 3 - 4 + -- - + /?(« + 1) = /|('l+1X"+2> for every positive integer n.
6.49. Prove that 4" > « 3 for every positive integer n.
6.50. Prove that 24 | (52" — 1) for every positive integer n.
6.51. By Result 6.5,
1 6 8 Chapter 6 M athem atical Induction
for every positive integer n .
(a) Use ( 6 . 1 2 ) to determine a formula for 2 2 + 42 + 6 2 H--- b ( 2k) 2 for every positive integer n.
(b) Use (6.12) and (a) to determine a formula for l 2 + 32 + 52 H--- f ( 2 n - l ) 2 for every positive integer/;.
(c) Use (a) and (b) to determine a formula for
l 2 — 2 2 + 32 — 42 H--- b ( - i ) " +1B2 for every positive integer n.
(d) Use mathematical induction to verify the formulas in (b) and (c).
6.52. Use the Strong Principle of M athem atical Induction to prove that for each integer n > 28, there are nonnegative integers x and y such that n = 5x + 8y.
6.53. Find a positive integer m such that for each integer n > m, there are positive integers x and y such that n = 3x + 5y. Use the Principle of M athem atical Induction to prove this.
6.54. Find a positive integer m such that for each integer n > m, there are integers x , y > 2 such that n = 2x + 3y. Use the Principle of M athem atical Induction to prove this.
6.55. A sequence {a,,} of real numbers is defined recursively by <•/, = 1. « 2 ~ 2 and a„ = £ " = i (*’ - l)^r for n > 3. Prove that a„ = (n - 1)! for every integer n > 3.
6.56. Consider the sequence a\ = 2, a 2 = 5, a3 = 9, = 14, etc.
(a) Find a recurrence relation that expresses a„ in terms of for every integer n > 2.
(b) Conjecture an explicit formula for a„, and then prove that your conjecture is correct.
6.57. The following theorem allows one to prove certain quantified statements over some finite sets.
T he P rinciple of F inite In d u ctio n For a fixed positive integer m, let S = { 1 ,2 ...m}. For each n e S let P(n) be a statement. I f
(a) P ( l ) is true and (b) the implication
I f P(k), then P (k + 1).
is true f o r every integer k with 1 < k < m, then P(n ) is true fo r every integer n e S.
Use the Principle of Finite Induction to prove the following result.
Let 5 = {1, 2 , . . ., 24}. For every integer t with 1 < t < 300, there exists a subset S, c S such that YieS, i =
f-6.58. Evaluate the proposed proof of the following result.
R esult For every positive integer n ,
1 + 3 + 5 + • • ■ + (2n - 1) = ir .
P ro o f We proceed by induction. Since 2 • 1 — 1 = l 2, the formula holds for n = 1. Assume that 1 + 3 + 5 + • ■ ■ + (2k — 1) = k2 for a positive integer k. We prove that 1 + 3 + 5 + ■ • • + (2k + 1) = (k + l) 2. Observe that
1 + 3 + 5 + • ■ • + (2k + 1) = (k + i f l + 3 + 5 + -- - + ( 2 * - l ) + ( 2 * + 1) = (k + l) 2
k2 + (2k + 1) = (k + I)2 0.- + 1)2 = ( * + l)2.
6.59. Below is given a proof of a result. W hich result is being proved and which proof technique is being used?
Additional Exercises for Chapter 6 169
P r o o f Assume, to the contrary, that there is some positive integer n such that 8 / ( 32,1 — 1). Let m be the sm allest positive integer such that 8 / (32m — 1). For n = 1, 32" — 1 = 8. Since 8 | 8, it follows that m > 2.
Let m = * + 1. Since 1 < * < m, it follows that 8 | (32k — 1). Therefore, 32* — 1 = 8x for some integer x and so 32k = 8x + 1. Hence
32"' - 1 = 32(ir+1) - 1 = 32*+2 - 1 = 9 • 32* - 1
= 9(8* + 1) - 1 = 72x + 8 = 8(9x + 1).
Since 9x + 1 is an integer, 8 | (32m — 1), which produces a contradiction. ■ 6.60. Below is given a proof of a result. Which result is being proved and which proof technique is being used?
P r o o f First observe that a\ = 8 = 3 • 1 + 5 and «2 — 1 1 = 3 - 2 + 5. Thus a„ = 3n + 5 for n = 1 and n = 2. Assume that a, = 3/ + 5 for all integers i with 1 < i < k, where k > 2 . Since k + 1 > 3, it follows that
cik+\ — 5ctk •— 4<7£_i — 9 = 5(3k + 5) — 4(3k + 2) — 9
= 15* + 25 - 12* - 8 - 9 = 3k + 8 = 3(* + 1) + 5. ■ 6.61. By an «-gon, we mean an w-sided polygon. So a 3-gon is a triangle and a 4-gon is a quadrilateral. It is well
known that the sum o f the interior angles of a triangle is 180°. Use induction to prove that for every integer n > 3, the sum of the interior angles o f an /?-gon is (n — 2) • 180°.
■?.62. Suppose that {an} is a sequence o f real numbers defined recursively by a\ = 1, <22 = 2, #3 = 3 and an = 2an- \ — a„ - 3 for n > 4. Prove that a„ = a„^\ + a„ - 2 for every integer n > 3.
5.63. Suppose that {«„} is a sequence of real numbers defined recursively by a\ = 1, a.2 = 2 and a„ = a „ _ i/a „_2 for n > 3.
(a) Prove that
1 if n = 1 ,4 (mod 6) 2 if n = 2, 3 (mod 6) 1 /2 if n = 0 ,5 (mod 6) for every positive integer n.
(b) Prove, for each nonnegative integer j , that Y^i= 1 ai+> = ^ ■
6.64. Let x e R where x > 3. Prove that (1 + x)n > [ ”("~ 1g('1~2:>] x 3 for every integer n > 3.
-.65. Prove that ^ X )/= i!) — '’("+l^ ni:2- for every positive integer n.
6.66. Prove that YTj= 1 ( S / = 1 (2/ — 1)^) = »(»+iX2«+i) for every positive integer n.
6.67. Prove that there exists an odd integer m such that every odd integer n with n > m can be expressed as 3a + 5b + 7c for positive integers a , b and c.