Assignment #6 (The same questions for both sessions)
Due Tuesday, 3/3
Complete the following exercises from chapter 2.4:
ł 2, 6
Complete the following exercises from chapter 4.1:
� 4, 6, 7, 8, 10, 20, 22, 28
Chapter 2.4: Exercise 2:
Answer:
Chapter 2.4: Exercise 6:
Answer:
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Chapter 4.1: Exercise 4: Let P(n) be the statement that 1
3
+ 2
3
+ 3
3
+ … + n
3
= (n(n+1)/2)
2
for the positive
integer n
a)
What is statement of P(1)?
Answer: 1
3
= (1(1+1)/2)
2
b)
Show that P(1) is true by completing the basis step of the proof
Answer: we show P(1) by showing both sides of P(1) are equal.
1
3
= (1(1+1)/2)
2
1
= 1
c)
What is inductive hypothesis?
Answer:
d)
What do you need to prove in the inductive step?
Answer: For each k
≥
1, P(k) implies P(k+1).
In other words, assuming the inductive
hypothesis, we want to show
e)
Complete the inductive step:
Answer:
[
]
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29 (
29
(
29
(
29 (
29
2
2
2
2
2
2
2
2
2
1
2
2
3
2
2
3
2
3
3
3
3
2
2
1
2
2
1
4
2
1
4
2
1
4
4
4
1
1
4
1
1
4
1
Hypothesis
Inductive
by
1
2
1
1
2
1
+
+
=
+
+
=
+
+
=
+
+
=
+
+
+
=
+
+
+
=
+
+
+
=
+
+
+
=
+
+
+
+
+
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
f)
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 Spring '09
 Mathematical Induction, Natural number, Peano axioms, 4 K, 2 K, 3 5k

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