CSC236H: Introduction to the Theory of Computatoin
Homework 1 Solutions
1. Use induction to prove that the following equation holds for all positive integers
n
:
n
X
k
=1
1
k
(
k
+ 1)
=
n
n
+ 1
.
Solution.
We prove this by induction on
n
.
Base case: Note that for
n
= 1 both sides of the equation equal 1
/
2.
Inductive hypothesis: Let
m
∈
N
and suppose the statement is true for
n
=
m

1, i.e. suppose that
m

1
X
k
=1
1
k
(
k
+ 1)
=
m

1
m
.
Induction step: Using the inductive hypothesis we have
m
X
k
=1
1
k
(
k
+ 1)
=
m

1
X
k
=1
1
k
(
k
+ 1)
+
1
m
(
m
+ 1)
=
m

1
m
+
1
m
(
m
+ 1)
=
(
m

1)(
m
+ 1) + 1
m
(
m
+ 1)
=
m
m
+ 1
.
Therefore the statement is also true for
n
=
m
.
2. Use induction to prove 3
n
< n
! for all
n >
6 with
n
∈
N
.
Solution.
Base case: The claim is made for
n >
6. Therefore the base case we need to verify is
n
= 7.
Since 3
7
= 2187 and 7! = 5040 the statement holds for the base case.
Inductive hypothesis: Let
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 Spring '10
 FarzanAzadeh
 Mathematical Induction, Cos, Natural number, inductive hypothesis, 1 1 k

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