N=2p-3, p€n and n² ÷ 4 =?
Answers
Step-by-step explanation:
P(1) is true since
1 · 2
2
2
=
2
2
= 1 = 13
.
(c) The inductive hypothesis is P(n):
1
3 + 23 + . . . + n
3 =
n(n + 1)
2
2
.
(d) Assuming the inductive hypothesis, we aim to show P(n + 1):
1
3 + 23 + . . . + n
3 + (n + 1)3 =
(n + 1)(n + 2)
2
2
.
(e) Suppose n is such that
1
3 + 23 + . . . + n
3 =
n(n + 1)
2
2
.
Then by our inductive hypothesis, we have
1
3 + . . . + n
3 + (n + 1)3 =
n(n + 1)
2
2
+ (n + 1)3
.
Now we compute
n(n + 1)
2
2
+ (n + 1)3 =
n
2
(n + 1)2
4
+ (n + 1)3
=
n
2
(n + 1)2
4
+
4(n + 1)3
4
=
(n
2 + 4(n + 1))(n + 1)2
4
=
(n
2 + 4n + 4)(n + 1)2
4
=
(n + 2)2
(n + 1)2
4
=
(n + 1)(n + 2)
2
2
.
This shows
1
3 + 23 + . . . + (n + 1)3 =
(n + 1)(n + 2)
2
2
.
(f) We showed P(1) and (∀n)(P(n) → P(n + 1)) by universal generalization. It follows by mathematical induction that
(∀n)P(n).
Exercise 1.2. 10
Proof. Notice we have
1
1 · 2
=
1
2
1
1 · 2
+
1
2 · 3
=
3
2 · 3
+
1
2 · 3
=
4
6
=
2
3
1
1 · 2
+
1
2 · 3
+
1
3 · 4
=
3 · 4
2 · 3 · 4
+
4
2 · 3 · 4
+
2
2 · 3 · 4
=
18
24
=
3
4
.