MACM 101 — Discrete Mathematics I
Outline Solutions to Exercises on Induction and Combinatorics
1.
Prove that for every positive integer
n
1
·
2
1
+ 2
·
2
2
+ 3
·
2
3
+
...
+
n
·
2
n
= (
n

1)2
n
+1
+ 2
.
We use induction. Let
P
(
n
)
denote this equality for the integer
n
.
Basis case.
P
(1)
means the equality
1
·
2
1
= (1

1)2
n
+1
+ 2
, which is obviously true.
Inductive step.
Suppose that
P
(
k
)
is true, that is,
1
·
2
1
+ 2
·
2
2
+ 3
·
2
3
+
...
+
k
·
2
k
= (
k

1)2
k
+1
+ 2
.
We have to prove
P
(
k
+ 1)
:
1
·
2
1
+ 2
·
2
2
+ 3
·
2
3
+
...
+
k
·
2
k
+ (
k
+ 1)
·
2
k
+1
=
k
2
k
+2
+ 2
.
We have
1
·
2
1
+ 2
·
2
2
+ 3
·
2
3
+
...
+
k
·
2
k
+ (
k
+ 1)
·
2
k
+1
=
(
k

1)
·
2
k
+1
+ 2 + (
k
+ 1)
·
2
k
+1
=
2
k
·
2
k
+1
+ 2
=
k
·
2
k
+2
+ 2
2.
Suppose there are
n
people in a group, each aware of a scandal no one else in the group knows about.
These people communicate by telephone; when two people in the group talk, they share information about
all scandals each knows about. The gossip problem asks for
G
(
n
)
, the minimum number of telephone calls
that are needed for all
n
people to learn about all the scandals.
Prove that
G
(
n
)
≤
2
n

3
.
Denote by
P
(
n
)
the cstatement that
G
(
n
)
≤
2
n

3
.
Basis step.
We prove
P
(2)
. If there are 2 people, then clearly they need only one phone call to exchange all the
gossips. Thus
G
(2) = 1
and
2
·
2

3 = 1
, hence,
G
(2)
≤
2
·
2

3
.
Inductive step.
Suppose that
P
(
k
)
is true, that is, a collection of
k
people needs at most
2
k

3
phone calls to
share all their gossips. The protocol for
k
+ 1
people goes as follows. The person number
k
+1
makes a phone
call to a person, say number
k
, and thus this
k
th person now know the gossip of the person number
k
+1
. Then
the first
k
people make at most
2
k

3
calls and share all their gossips including the gossip from the person
number
k
+ 1
. After that everyone except for
(
k
+ 1)
th person knows all the gossips. Now the person number
k
+ 1
makes a call again to anyone and learns the gossips.
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 Spring '09
 jcliu
 Mathematical Induction, Natural number, inductive step, complete binary tree, basis step

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