1.2 2· 3 Prove the following statement by mathematical induction. For every integer n z 1, + +... %3D 3.4 n(n + 1) n + 1 Proof (by mathematical induction): Let P(n) be the equation 1 +... + %3D 1.2 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. 1. 2 1 + 1 1 O P(1) = 1. 2 O P(1) = 1+ 1 1: 2 1(1 + 1) 1 +1 1 1 1 + %3D 1.2 2:3 3. 4 1: 2 1 1 +1 The selected statement is true because both sides of the equation equal the same quantity. Show that for each integer k 2 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k > 1, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below. 1 1:2 k(k + 1) 1 1 1.2 2.3 3.4 1. 1:2 2: 3 3.4 k(k + 1) 1 1. + 1. 2 2.3 3. 4 1 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 1·2 2:3 3. 4 + 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 k + 1 k(k + 1) (k + 1)(k + 2) 1 1. 1.2 k(k + 1) (k + 1)(k + 2) 1 1. 1. + 1: 2 2.3 3. 4 k(k + 1) 1 (k + 1)(k + 2) When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal · Hence P(k + 1) is true, which completes the inductive step. [Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.] Need Help? Read It
1.2 2· 3 Prove the following statement by mathematical induction. For every integer n z 1, + +... %3D 3.4 n(n + 1) n + 1 Proof (by mathematical induction): Let P(n) be the equation 1 +... + %3D 1.2 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. 1. 2 1 + 1 1 O P(1) = 1. 2 O P(1) = 1+ 1 1: 2 1(1 + 1) 1 +1 1 1 1 + %3D 1.2 2:3 3. 4 1: 2 1 1 +1 The selected statement is true because both sides of the equation equal the same quantity. Show that for each integer k 2 1, if P(k) is true, then P(k + 1) is true: Let k be any integer with k > 1, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below. 1 1:2 k(k + 1) 1 1 1.2 2.3 3.4 1. 1:2 2: 3 3.4 k(k + 1) 1 1. + 1. 2 2.3 3. 4 1 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 1·2 2:3 3. 4 + 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 k + 1 k(k + 1) (k + 1)(k + 2) 1 1. 1.2 k(k + 1) (k + 1)(k + 2) 1 1. 1. + 1: 2 2.3 3. 4 k(k + 1) 1 (k + 1)(k + 2) When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal · Hence P(k + 1) is true, which completes the inductive step. [Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.] Need Help? Read It
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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Question
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![Prove the following statement by mathematical induction.
For every integer n 2 1,
+
1: 2
3.4
n(n + 1)
n + 1
Proof (by mathematical induction): Let P(n) be the equation
1
1
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.
1
1
1:2
1 + 1
O P(1) =
O P(1) =
1 + 1
1: 2
1(1 + 1)
1 + 1
+
1: 2
2: 3
+
3. 4
1: 2
1
1 + 1
The selected statement is true because both sides of the equation equal the same quantity.
Show that for each integer k2 1, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k > 1, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.
1.
lo
1
+
1:2
k(k + 1)
1.
1
1: 2
2:3
3. 4
1
1
1
+
1:2
2: 3
3. 4
k(k + 1)
1
+
1:2
2: 3
3. 4
1
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
1:2
2: 3
3. 4
+ 1
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
1:2
k(k + 1)
(k + 1)(k + 2)
1.
1
1
+
1·2
2·3
3. 4
k(k + 1)
(k + 1)(k + 2)
When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal
· Hence P(k + 1) is true, which completes the inductive step.
[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]
Need Help?
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Transcribed Image Text:Prove the following statement by mathematical induction.
For every integer n 2 1,
+
1: 2
3.4
n(n + 1)
n + 1
Proof (by mathematical induction): Let P(n) be the equation
1
1
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.
1
1
1:2
1 + 1
O P(1) =
O P(1) =
1 + 1
1: 2
1(1 + 1)
1 + 1
+
1: 2
2: 3
+
3. 4
1: 2
1
1 + 1
The selected statement is true because both sides of the equation equal the same quantity.
Show that for each integer k2 1, if P(k) is true, then P(k + 1) is true:
Let k be any integer with k > 1, and suppose that P(k) is true. Select the expression for the left-hand side of P(k) from the choices below.
1.
lo
1
+
1:2
k(k + 1)
1.
1
1: 2
2:3
3. 4
1
1
1
+
1:2
2: 3
3. 4
k(k + 1)
1
+
1:2
2: 3
3. 4
1
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
1:2
2: 3
3. 4
+ 1
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
1:2
k(k + 1)
(k + 1)(k + 2)
1.
1
1
+
1·2
2·3
3. 4
k(k + 1)
(k + 1)(k + 2)
When the left-hand and right-hand sides of P(k + 1) are simplified, they both can be shown to equal
· Hence P(k + 1) is true, which completes the inductive step.
[Thus both the basis and the inductive steps have been proved, and so the proof by mathematical induction is complete.]
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Read It
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