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

136 The Nuts and Bolts of Proof, Third Edition
4. a. We have to check whether the statement holds true for /c = 3:
(1 + a)^ = 1 + 3fl + 3^2 + a^ > 1 + 3al
The inequaUty is true because all the numbers used are positive.
b. Let us assume that the inequality is true for k = n\ that is, assume
that {\ + af>\ + na^.
c. Let us check whether (1 + a)""^^ >! + («+ l)a^. We can use rules
of algebra and the inductive hypothesis to obtain
Because a > 1, a^>a^. So
(l + ar+i=(H-«)"(! +a)
> (1 + na^){\ + a)
= l-{-na^ -\-a-\-na^
> l-\- a^ -\- na^.
(1 + a)"+^ > 1 + na^ + na^
> 1 + na^ + na^
> 1 + na^ + a^
= l-\-in-\-l)a\
Thus, by the principle of mathematical induction, the original
statement is true.
5. a. We will check the equahty for fc = L In this case, the left-hand side of
the equation has only one term: 1/2. The right-hand side is equal to:
- i '•-
So the equality is true for /c= 1.
b. Let us assume the equahty holds true for k = n.
c. We will try to prove that
2 + '""^V2/ W l-(l/2)