Math 320-101 CRN 22189
Spring 2016
Test 1
Carter
1. In an implication P ⇒ Q, which of P , Q is called the antecedent? Which is called
the consequent?
2. Suppose that 0 < d and that d, n ∈ Z. Define d|n.
3. The statement, “Every non-empty set of natural numbers has a least element.” is called
the
.
4. Describe the two steps used to prove something by means of mathematical induction.
.
(a) First prove
(b) Then prove
.
5. Suppose that 3 divides both a and b.
(a) Prove that for any integers x and y, the quantity xa + by is divisible by 3.
(b) Prove that a · b is also divisible by 3.
6. Prove that any non-zero integer is divisible by itself.
7. Suppose that a, b, c ∈ Z. For convenience assume that a, b, c 6= 0. Suppose that a
divides b and that b divides c. Show that a divides c.
8. Let r ∈ R be an arbitrary real number r 6= 1. Prove by induction that
1 + r + r2 + · · · + rn = (rn+1 − 1)/(r − 1).
9. Prove by induction that
n
X
k=1
(2k − 1) = k 2 .