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

108 The Nuts and Bolts of Proofs
5. The reciprocal of a nonzero number of the kind z = a + bVS, with a and
b real numbers, is a number of the same kind.
6. If X is a positive real number, then x^ > x.
7. Let n be a natural number and x a fixed positive irrational number. Then
y/x is always irrational.
8. Let n be an odd number. Then n{n^ — 1) is divisible by 24.
9. Let/be a differentiable increasing function. Then -^/(x) is an increasing
function. (*)
10. Let a, fo, c, and n be four positive integers. The numbers a, b, and c are
divisible by n if and only if a + & + c is divisible by n.
11. Consider the equation ax^ + bx + c = 0, with a^O, and fe^ — 4ac > 0.
^, . , . -b + Vb^ - 4ac ^ -fe - Vfc2 - 4ac
Then its solutions are xi = and X2 = .
2a
la
12. Let a and b be any two real numbers. Prove the following statements:
(a) {a-^bf>Aab
(b) {a + bf = Aab if and only if a = ft.
13. The sum of two consecutive numbers is divisible by 2.
14. Let a and d be two fixed positive integer numbers. Then
a + (a + rf) + (a + 2rf) + (a + 3d) + .... 4- (a + nd) = (^+^X^^ + ^^)
for all integers n> 1.
15. Let n be a natural number. Then n is a multiple of 7 if and only if n^ is a
multiple of 7.
16. Let/(x) = 15x + 7. Then lim/(x) = 37.
x->2
17. The product of two consecutive numbers is divisible by 2.
18. If a, fc, and c are three integers such that a^ + b^ = c^ {i.e., they are a
Pythagorean triple), then they cannot all be odd.
19. The square of an odd integer is a number of the form 8t + 1, where t is
an integer.
20. Let n be a number that is not a multiple of 3. Then either n + 1 or n — 1 is
a multiple of 3.
21. If (a + bf= a^ + fc^ then either a = OoTb = 0.