Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2. For every integer n 21, 1+6+11+16++ (Sn-4)= n(Sn - 3) 2 Proof (by mathematical induction): Let P(n) be the equation 1+6+ 11 + 16 + + (Sn-4)= n(Sn-3) 2 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. 01-1-(5-1-3) 2 O P(1) = 1-(5-1-3) 2 1+(5-1-4)=1-(5-1-3) O P(1) = 5-1-4 The selected statement is true because both sides of the equation equal Show that for each integer k 21, 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. The left-hand side of P(k) is-Select-- 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 + 6 + 11 + 16 ++ (5(k+ 1) - 4) = -Select- ✓, and the right-hand side of P(k) is After substitution from the inductive hypothesis, the left-hand side of P(k+ 1) becomes Hence P(k+ 1) is true, which completes the inductive step +(5(k+1)-4). 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.]
Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2. For every integer n 21, 1+6+11+16++ (Sn-4)= n(Sn - 3) 2 Proof (by mathematical induction): Let P(n) be the equation 1+6+ 11 + 16 + + (Sn-4)= n(Sn-3) 2 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. 01-1-(5-1-3) 2 O P(1) = 1-(5-1-3) 2 1+(5-1-4)=1-(5-1-3) O P(1) = 5-1-4 The selected statement is true because both sides of the equation equal Show that for each integer k 21, 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. The left-hand side of P(k) is-Select-- 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 + 6 + 11 + 16 ++ (5(k+ 1) - 4) = -Select- ✓, and the right-hand side of P(k) is After substitution from the inductive hypothesis, the left-hand side of P(k+ 1) becomes Hence P(k+ 1) is true, which completes the inductive step +(5(k+1)-4). 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.]
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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![Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2.
For every integer n ≥ 1, 1+ 6 + 11 + 16 +
+ (5n - 4) = n(5n − 3)
2
Proof (by mathematical induction): Let P(n) be the equation
n(5n - 3)
2
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-(5-1-3)
2
01=
1 +6+11+ 16 + + (5n - 4) =
=
1-(5-1-3)
2
Ⓒ1+(5-1-4)=1-(5-1-3)
O P(1) = 5-1-4
The selected statement is true because both sides of the equation equal
Show that for each integer k ≥ 1, if P(k) is true, then P(k + 1) is true:
O P(1) =
Let k be any integer with k21, and suppose that P(k) is true. The left-hand side of P(k) is ---Select---
[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 + 6 + 11 + 16 + ... + (5(k+ 1) − 4) =
---Select--
and the right-hand side of P(k) is
After substitution from the inductive hypothesis, the left-hand side of P(k+ 1) becomes
+ (5(k+1)-4). 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.]
Hence P(k+ 1) is true, which completes the inductive step.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa0e38307-1ade-44bc-b712-aaeda4c58098%2F06a9b12c-1fe0-4f37-9875-7ea3b9b1fd51%2Fl957i7j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Prove the following statement using mathematical induction. Do not derive it from Theorem 5.2.1 or Theorem 5.2.2.
For every integer n ≥ 1, 1+ 6 + 11 + 16 +
+ (5n - 4) = n(5n − 3)
2
Proof (by mathematical induction): Let P(n) be the equation
n(5n - 3)
2
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-(5-1-3)
2
01=
1 +6+11+ 16 + + (5n - 4) =
=
1-(5-1-3)
2
Ⓒ1+(5-1-4)=1-(5-1-3)
O P(1) = 5-1-4
The selected statement is true because both sides of the equation equal
Show that for each integer k ≥ 1, if P(k) is true, then P(k + 1) is true:
O P(1) =
Let k be any integer with k21, and suppose that P(k) is true. The left-hand side of P(k) is ---Select---
[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 + 6 + 11 + 16 + ... + (5(k+ 1) − 4) =
---Select--
and the right-hand side of P(k) is
After substitution from the inductive hypothesis, the left-hand side of P(k+ 1) becomes
+ (5(k+1)-4). 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.]
Hence P(k+ 1) is true, which completes the inductive step.
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