Assignment 1, Math 220
Due: Friday, January 18th, 2002
1 Conjecture a formula for
n
X
1
1
1
1
=
+
+···+
k(k + 1)
1·2 2·3
n · (n + 1)
k=1
from the value of this sum for small integers n. Prove your conjecture
using mathematical induction.
2 Use mathematical induction to prove Bernoulli’s inequality : If
1 + x > 0, then
(1 + x)n ≥ 1 + nx,
for all n ∈ N.
3 Show that any amount of postage that is an integer number greater
than 11 cents can be formed using just 4-cent and 5-cent stamps.
4 Using our axioms for R, prove the the following assertions :
a: If x ∈ R, then −(−x) = x.
b: If x and y are real numbers of the same sign (x and y are either
both positive or both negative), and x < y, then 1/y < 1/x.
c: If x is a real number with x 6= 0 then x2 > 0.
5 Prove that there is no n ∈ N such that 0 < n < 1 (Use the WellOrdering Property of N).
1