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'

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

Pls. provide complete explanation and right answer.

Answer no. 6,7,8,9,10 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'
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'
Expert Solution
steps

Step by step

Solved in 2 steps with 1 images

Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,