1. 5" – 1 is divisible by 4, Vn E W. 2. 8" – 3" is divisible by 5, Vn E W. 3. x2n – y2n has a factor x + y, Vn E N. - 4. n? < n!, for any integer n > 4 5. (a+1)" > 1+ na, where a > -1, Vn € N
1. 5" – 1 is divisible by 4, Vn E W. 2. 8" – 3" is divisible by 5, Vn E W. 3. x2n – y2n has a factor x + y, Vn E N. - 4. n? < n!, for any integer n > 4 5. (a+1)" > 1+ na, where a > -1, Vn € N
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
Pls. provide complete explanation and right answer.
Answer no. 1,2,3,4,5 only
![IV. Prove the following using mathematical induction.
1. 5" – 1 is divisible by 4, Vn € W.
2. 8" – 3" is divisible by 5, Vn E W.
3. x2n – y2n has a factor x +y, Vn E N.
-
4. n? < n!, for any integer n > 4
5. (a + 1)" >1+ na, where a > -1, Vn E N
1
6. 3+32 + 3³ +...+ 3ª =
(3n+1 – 3), Vn E N.
2
|
7. 12 + 32 + 52 + ...+ (2n – 1)? = n(2n – 1)(2n +1)
Vn E N.
|
3
(2n – 1)3n+1 + 3
8. 1.3+2.32 + 3 · 33 + ... + n · 3ª =
Vn E N.
4
n°(n+1)²
9. 13 + 23 + 33 + ...+ n³
Vn E N.
%3D
4
1
10.
1.2
1
1
n
+.
3.4
...
2-3
n. (n+1)
n+1'](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb7c9b0fe-bcaa-47ab-9169-4992688d428b%2Fd4540032-2aa5-4b89-aa40-a100566a7dd9%2F5tsktb_processed.png&w=3840&q=75)
Transcribed Image Text:IV. Prove the following using mathematical induction.
1. 5" – 1 is divisible by 4, Vn € W.
2. 8" – 3" is divisible by 5, Vn E W.
3. x2n – y2n has a factor x +y, Vn E N.
-
4. n? < n!, for any integer n > 4
5. (a + 1)" >1+ na, where a > -1, Vn E N
1
6. 3+32 + 3³ +...+ 3ª =
(3n+1 – 3), Vn E N.
2
|
7. 12 + 32 + 52 + ...+ (2n – 1)? = n(2n – 1)(2n +1)
Vn E N.
|
3
(2n – 1)3n+1 + 3
8. 1.3+2.32 + 3 · 33 + ... + n · 3ª =
Vn E N.
4
n°(n+1)²
9. 13 + 23 + 33 + ...+ n³
Vn E N.
%3D
4
1
10.
1.2
1
1
n
+.
3.4
...
2-3
n. (n+1)
n+1'
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