COT3100C01, Fall 2002
S. Lang
Solution Keys to Assignment #4 (40 pts.)
10/24/2002
Use
induction
to prove each of the following questions, and be sure to
mark clearly when and
where the induction hypothesis is applied in each question:
1.
(10 pts.)
.
16
5
)
1
4
(
5
5
,
1
integer
for
that
Prove
1
1
+

=
∑
≥
+
=
n
j
n
n
n
j
j
Answer
: We use induction on
n
≥
1.
(Basis Step) Consider
n
= 1.
In this case, the LHS =
.
5
5
1
5
1
1
1
=
⋅
=
∑
=
j
j
j
The RHS =
.
5
16
80
16
5
3
25
16
5
)
1
4
(
5
1
1
=
=
+
⋅
=
+

+
Thus, LHS = RHS, so the Basis Step is proved
(Induction Hypothesis) Consider
n
=
k
.
Suppose
16
5
)
1
4
(
5
5
1
1
+

=
∑
+
=
k
j
k
k
j
j
for some
k
≥
1.
(Induction Step) Consider
n
=
k
+1.
We need to prove
(1)



16
5
)
3
4
(
5
16
5
)
1
)
1
(
4
(
5
5
2
1
1
1
1
+
+
=
+

+
=
∑
+
+
+
+
=
k
k
j
k
k
k
j
j
(1).
of
RHS
16
5
)
3
4
(
5
16
5
)
3
4
(
5
5
16
5
)
15
20
(
5
16
5
)
16
16
1
4
(
5
16
5
)
1
(
16
5
)
1
4
(
5
Hypothesis
Induction
by the
,
5
)
1
(
16
5
)
1
4
(
5
summation
of
definition
by the
,
5
)
1
(
5
(1)
of
LHS
that the
Note
2
1
1
1
k
1
1
k
1
1
k
1
1
=
+
+
=
+
+
⋅
=
+
+
=
+
+
+

=
+
+
+

=
+
+
+

=
+
+
∑
=
+
+
+
+
+
+
+
+
+
=
k
k
k
k
k
k
k
k
k
k
j
k
k
k
k
k
k
k
j
j
Thus, the Induction Step is proved.
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 Spring '09
 Mathematical Induction, Recursion, Inductive Reasoning, Natural number, Structural induction

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