Download Problem Set #1

Problem Set # 1 / MATH 16100
Due: Tuesday, October 6, 2015, at 11:50 CDST
Note 1: This problem set is intended to help you get comfortable with writing proofs by induction.
Note 2: In these problems, you may assume that we have defined addition, multiplication, and order on
the real numbers in the usual way, so that all of the usual properties hold.
1. Prove that, for every n ∈ N,
1 2 + 2 2 + · · · + n2 =
1
n (n + 1) (2n + 1) .
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Remark: We often use the sigma notation to express a sum like the one on the left-hand side of the
equation above. We write
n
X
j 2 := 12 + 22 + · · · + n2 .
j=1
The symbol on the left-hand side of this equation is verbalized as “the sum of j 2 , from j = 1 to j = n”.
The symbol “:=” means “is defined by”. The character j is called an index of summation and has no
meaning outside of the Σ summation notation. Thus, you can describe the same sum by using different
characters, e.g.,
n
n
n
X
X
X
12 + 22 + · · · + n2 =
j2 =
k2 =
z2 .
j=1
k=1
z=1
The most commonly-used indices of notation are i, j, k, `, m, and n. You should probably choose from
these unless there is a compelling reason to avoid all of them.
2. Prove that, for every natural number n ≥ 4,
n2 ≤ 2n .
3. Prove Bernoulli’s inequality: For every n ∈ N and every real number x > −1,
n
1 + nx ≤ (1 + x) .
4. Have you ever seen two horses of different colors from each other? Criticize the following “proof” that,
among every set of horses, all of the horses in the set are the same color.
Proof. For each n ∈ N, let P (n) be the mathematical statement that every set of n horses satisfies
the property that all of the horses in the set are the same color. We will prove that P (n) is true for
each n ∈ N by mathematical induction. First, for the base case, if n = 1, it is clear that P (n) holds,
as there can only be one color. Now, suppose P (n) is true for some n ∈ N. We need to prove that
P (n + 1) must also be true. Thus, suppose you are given a set H of n + 1 horses. Let any two horses,
h1 and h2 , be selected arbitrarily from H. First, remove h1 from H to leave yourself with a new set
H1 of n horses, one of which is h2 . By the induction hypothesis, the horses in H1 are all the same
color. Now put h1 back into the set to reform H and remove h2 from H to leave yourself with another
new set H2 of n horses, one of which is h1 . By the induction hypothesis again, the horses in H2 are
also all the same color. It follows from this that h1 and h2 are the same color, and thus, since they
were selected arbitrarily from our set H, all of the horses in H are the same color. This proves that
P (n + 1) is true and so, by mathematical induction, P (n) is true for all n ∈ N. That is, among every
set of horses, all of the horses in the set are the same color.
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