We must show that P(k + 1) is true. P(k + 1) is the equation 1 1 +... + + 3. 4 k(k + 1) + 1 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 1 1· 2 k(k + 1) (k + 1)(k + 2) 1 1 1 1 1: 2 2:3 3. 4 k(k + 1) (k + 1)(k + 2)
We must show that P(k + 1) is true. P(k + 1) is the equation 1 1 +... + + 3. 4 k(k + 1) + 1 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 1 1· 2 k(k + 1) (k + 1)(k + 2) 1 1 1 1 1: 2 2:3 3. 4 k(k + 1) (k + 1)(k + 2)
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 49E: Show that if the statement
is assumed to be true for , then it can be proved to be true for . Is...
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Transcribed Image Text:Prove the following statement by mathematical induction.
1
For every integer n 2 1,
1
1
1
n
+
+...
1. 2
2:3
3. 4
n(n + 1)
n + 1
Proof (by mathematical induction): Let P(n) be the equation
1
1
1
- +. .. +
1
1.2
3. 4
n(n + 1)
n + 1
2:3
We will show that P(n) is true for every integer n > 1.
Show that P(1) is true: Select P(1) from the choices below.
1
1
+ –
1
+ -
1
+ -
1. 2
2:3
3. 4
1· 2
1
1 + 1
1
1
1
+
1. 2
1(1 + 1)
1 + 1
1
O P(1) :
1· 2
1
O P(1)
1 + 1
1
1
1
2
1 + 1
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 > 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
+
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
1
+
+
1.2
2: 3
3. 4
1
k(k + 1)
k
The right-hand side of P(k) is
k+1
[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
1
+... +
k(k + 1)
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
+
1· 2
k(k + 1)
(k + 1)(k + 2)
1
1
1
1
1
+
1. 2
2:3
3. 4
k(k + 1)
(k + 1)(k + 2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8268c271-8991-49a2-aed9-a02bde5bd8ab%2F49df4bc9-01bf-4841-94b4-ae2e033c5db9%2Fcwoxlcp_processed.png&w=3840&q=75)
Transcribed Image Text: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 > 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
+
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
1
+
+
1.2
2: 3
3. 4
1
k(k + 1)
k
The right-hand side of P(k) is
k+1
[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
1
+... +
k(k + 1)
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
+
1· 2
k(k + 1)
(k + 1)(k + 2)
1
1
1
1
1
+
1. 2
2:3
3. 4
k(k + 1)
(k + 1)(k + 2)
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