Homework 2, Math 245, Fall 2010
Due Wednesday, September 22 in class.
The solution of each exercise should be at most one page long. If you can, try to write
your solutions in LaTex. Each question is worth 2 points.
1. In a tournament with
n
≥
2 teams, every team plays against every other team exactly
once. Prove that the number of games played is
n
(
n

1)
2
.
Proof.
1st proof by induction
We will prove the statement by induction on
n
.
The base case is
n
= 2. The number of games in a tournament with 2 teams is 1 which
coincides with
2(2

1)
2
= 1. Thus, the base case is true.
The induction step. We assume that in any tournament with
k
teams, there are
k
(
k

1)
2
games played and we will prove that in any tournament with
k
+ 1 teams, there are
k
(
k
+1)
2
games played.
Consider an arbitrary tournament with
k
+ 1 teams. Assume the teams are called
T
1
,...,T
k
,T
k
+1
. One can think of the teams
T
1
,...,T
k
as forming a subtournament of
k
teams. By induction hypothesis, the number of games between the teams
T
1
,...,T
k
is
k
(
k

1)
2
. To ﬁnd the total number of games played in the tournament with
k
+1 teams,
we must add to
k
(
k

1)
2
the number of games that involve team
T
k
+1
. Team
T
k
+1
plays
k
games (one against each
T
i
for 1
≤
i
≤
k
). Thus, the number of games in the tournament
with
k
+ 1 teams is
k
(
k

1)
2
+
k
=
k
(
k
+1)
2
. This ﬁnishes our proof.
2nd proof without induction
In a tournament with
n
teams, each team plays
n

1
games as it has to play every other team exactly once. When we add up these numbers,
we double count each game exactly twice as every game involves exactly two teams.
This means that the total number of games played is
n
(
n

1)
/
2.
2. Find and prove a simple formula for the following sum:
1
·
x
+ 2
·
x
2
+
···
+
n
·
x
n
.
Proof.
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 Fall '10
 CIOABA
 Math, Mathematical Induction, 1 k, base case, Xn, Structural induction, 1 1 k

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