epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Exercises Without Solutions 111
(b) The function g, defined as g(x) = j ^ , is decreasing.
(c) The function /c„, defined as /c„(x) = nf(x), is increasing for all positive
numbers n.
50. The number 3" — 1 is divisible by 2 for all natural numbers n.
51. Let n be a positive multiple of 3, with n > 3. Then either n is a multiple of
6 or it is a multiple of 9.
52. For every counting number n, Yll=i ^=\/S^7^-
(Hint: Start by squaring both sides of the equality.)
53. There is a differentiable function/such that 0</(x)< 1 and)(0) = 0. (*)
54. If ab is divisible by 10, then either a or ft is divisible by 10.
55. Let/be a nonconstant function. Then/cannot be even and odd at the
same time.
56. Let a be a positive integer. If 3 does not divide a, then 3 divides a^ — I.
57. There exists a set of four consecutive integers such that the sum of the
cubes of the first three is equal to the cube of the largest number.
58. Let A, B, and C be three subsets of the same set U, Then
A - (Bn C) = (A - B)U{A - C).
59. A five-digit palindrome number is divisible by 11.
60. Let m and n be two integer numbers. Then the following statements are
equivalent:
(a) m and n are both odd numbers.
(b) mn is an odd number.
(c) m^n^ is an odd number.
61. There exist irrational numbers a and b such that a^ is an integer.
62. Let/(x) = ^/xTl. Then lim/(x) = 2.
{Hint: Use, in a suitable way, the conjugate of y/x-\-l — 2.)
63. Let / g, and h be three functions defined for all real numbers such
that h and g are increasing and h{x) <fix) < g{x) for all x. Then / is an
increasing function.
64. Let n be an integer number. The following statements are equivalent:
(a) n is divisible by 5.
(b) n^ is divisible by 25.
(c) n^ is divisible by 125.