Result For every integer n ≥ 5, 2">n². Proof We proceed by induction. Since 25 > 5², the inequality holds for n = 5. Assume that 2k > k² where k ≥ 5. We show that 2k+1 > (k+ 1)². Observe that 2k+1 = 2.2k > 2k² = k² + k² ≥ k² + 5k = k² + 2k + 3k ≥ k² + 2k + 15 > k² + 2k + 1 = (k+1)². Therefore, 2k+1> (k + 1)². By the Principle of Mathematical Induction, 2" >n² for every integer n ≥ 5.
Result For every integer n ≥ 5, 2">n². Proof We proceed by induction. Since 25 > 5², the inequality holds for n = 5. Assume that 2k > k² where k ≥ 5. We show that 2k+1 > (k+ 1)². Observe that 2k+1 = 2.2k > 2k² = k² + k² ≥ k² + 5k = k² + 2k + 3k ≥ k² + 2k + 15 > k² + 2k + 1 = (k+1)². Therefore, 2k+1> (k + 1)². By the Principle of Mathematical Induction, 2" >n² for every integer n ≥ 5.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
I understand the proof until we get to the highlighted part, please explain it in more steps, thank you in advance.
![Result
For every integer n ≥ 5,
2">n².
Proof We proceed by induction. Since 25 > 5², the inequality holds for n = 5. Assume that
2kk² where k≥ 5. We show that 2k+¹ > (k+ 1)². Observe that
2k+1
= 2.2k > 2k² = k² +k² > k² + 5k
= k² + 2k + 3k ≥k² + 2k +15
- k² + 2k + 1 = (k + 1)².
Therefore, 2k+¹ > (k+ 1)². By the Principle of Mathematical Induction, 2″ > n² for
every integer n ≥ 5.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8e237d3f-b8e6-4775-a6f9-5671b153aef2%2F418c0b38-380c-4e7d-963f-dc911f0ff6ca%2F73vqkc5_processed.png&w=3840&q=75)
Transcribed Image Text:Result
For every integer n ≥ 5,
2">n².
Proof We proceed by induction. Since 25 > 5², the inequality holds for n = 5. Assume that
2kk² where k≥ 5. We show that 2k+¹ > (k+ 1)². Observe that
2k+1
= 2.2k > 2k² = k² +k² > k² + 5k
= k² + 2k + 3k ≥k² + 2k +15
- k² + 2k + 1 = (k + 1)².
Therefore, 2k+¹ > (k+ 1)². By the Principle of Mathematical Induction, 2″ > n² for
every integer n ≥ 5.
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