For every integer n 2 1, 1.2 +.... 2-3 3.4 n(n + 1) n+1 Proof (by mathematical induction): Let P(n) be the equation 1 1 1. +...+ n(n + 1) %3D n+ 1 1-2 2.3 3.4 We will show that P(n) is true for every integer n2 1. Show that P(1) is true: Select P(1) from the choices below. O P(1) %3D 1 + 1 1 %3D 1-2 1(1 + 1) 1 + 1
For every integer n 2 1, 1.2 +.... 2-3 3.4 n(n + 1) n+1 Proof (by mathematical induction): Let P(n) be the equation 1 1 1. +...+ n(n + 1) %3D n+ 1 1-2 2.3 3.4 We will show that P(n) is true for every integer n2 1. Show that P(1) is true: Select P(1) from the choices below. O P(1) %3D 1 + 1 1 %3D 1-2 1(1 + 1) 1 + 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Prove the following statement by mathematical induction.
1
For every integer n 2 1,
1
1
in
1.2
2.3
3. 4
n(n + 1)
n+ 1
Proof (by mathematical induction): Let P(n) be the equation
1
1.2
2.3
3.4
n(n + 1)
n+ 1
We will show that P(n) is true for every integer n 2 1.
Show that P(1) is true: Select P(1) from the choices below.
O P(1) =
1
1+ 1
1(1 + 1) 1
1-2
3-4
1-2
1 +1
1- 2
O P(1) = 12
![The selected statement is true because both sides of the equation equal the same quantity.
Show that for each integer k> 1, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k 2 1, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.
1.2+
1-2
k(k + 1)
1
1
1- 2
2.3
3. 4
1
2.3 3.4 K(k + 1)
1
1
1
1-2
1
1
3 - 4
+
1- 2
2-3
k(k + 1)
The right-hand side of P(k) is
[The inductive hypothesis states that the two sides of P(k) are equal.]
We must show that P(k + 1) is true. P(k + 1) is the equation
1
1
3-4
1
%3!
+...+
1- 2
2-3
Which of the following choices shows the result of applying the inductive hypothesis to the expression on the left-hand side of P(k + 1)?
k
1.
k +1
(k + 1)(k + 2)
k
1
1
k + 1
k(k + 1)
(k + 1)(k + 2)
1
1
+
k(k + 1)
(k + 1)(k + 2)
1-2
1
3. 4
1
1
1. 2
2.3
k(k + 1)
(k + 1)(k + 2)
Hence P(k + 1) is true, which completes the inductive step.
When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal
[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99894a4a-cfa0-4bf4-9590-22770e9dd01d%2F6c422c05-62b0-46dc-a45b-ff754568d78b%2Fdr88qyh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The selected statement is true because both sides of the equation equal the same quantity.
Show that for each integer k> 1, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k 2 1, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.
1.2+
1-2
k(k + 1)
1
1
1- 2
2.3
3. 4
1
2.3 3.4 K(k + 1)
1
1
1
1-2
1
1
3 - 4
+
1- 2
2-3
k(k + 1)
The right-hand side of P(k) is
[The inductive hypothesis states that the two sides of P(k) are equal.]
We must show that P(k + 1) is true. P(k + 1) is the equation
1
1
3-4
1
%3!
+...+
1- 2
2-3
Which of the following choices shows the result of applying the inductive hypothesis to the expression on the left-hand side of P(k + 1)?
k
1.
k +1
(k + 1)(k + 2)
k
1
1
k + 1
k(k + 1)
(k + 1)(k + 2)
1
1
+
k(k + 1)
(k + 1)(k + 2)
1-2
1
3. 4
1
1
1. 2
2.3
k(k + 1)
(k + 1)(k + 2)
Hence P(k + 1) is true, which completes the inductive step.
When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal
[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]
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