Mathematical Induction

MATHEMATICAL INDUCTION:

Mathematical induction is a unique and special way to prove the things, in only two steps.

Step 1. Show that it is true for n = 1.
Step 2. Show that if n = k is true then n = k+1 is also true.


For example:

Prob. By principal of mathematical induction prove that
11n+2 + 122n+1 is divisible by 133, n ∈ N.

Solution.
Step 1 P(1)- Show it is true for n = 1
11n+2 + 122n+1 = 111+2 + 122(1)+1 = 1331 + 1728 = 3059

Yes 3059 is divisible by 133.
111+2 + 122(1)+1  is true.

Step 2 P(k)- Assume it is true for n = k

11k+2 + 122(k)+1  is true.
(above line is an assumption only, which we will use as a fact in rest of the solution)

Now, prove that 11(k+1)+2 + 122(k+1)+1 is divisible by 133. (here n = k +1 now, P(k+1))
We have,
P(k+1)

11(k+1)+2 + 122(k+1)+1 = 11k+3 + 122k+3

11(k+1)+2 + 122(k+1)+1 = 11k+2 x 11 + 122k+1x 122

11(k+1)+2 + 122(k+1)+1 = (11k+2 x 11) + (122k+1x 144 )

11(k+1)+2 + 122(k+1)+1 = (11k+2 x 11) + (122k+1x (11 + 133))

11(k+1)+2 + 122(k+1)+1 = (11k+2+ 122k+1)x 11+(122n+1 x 133)

11(k+1)+2 + 122(k+1)+1 = ((11k+2+122k+1)x 11)+(122n+1 x 133)

Here 11k+2+122k+1 is divisible by 133 as assumed in n = k, P(1),

And 122n+1 x 133 is multiple of 133 so it is divisible by 133.

So,

11(k+1)+2 + 122(k+1)+1 = ((divisible by 133)x 11)+(divisible by 133)

11(k+1)+2 + 122(k+1)+1 = divisible by 133.

In this problem

If n = n, i.e, P(1) is true then n = n+1, i.e, P(n+1) is also true. Hence proved.


Problems based on above study:


Prob. Prove the following proposition
12 + 22 + 32 +...........+ n2  = (n (n+1)(2n+1) ) /6.


Prob. By principal of mathematical induction prove that
11n+2 + 122n+! is divisible by 133, n ∈ N.


Prob. By mathematical induction prove that for any integer n,  11n+2 + 122n+1 is divisible by 133.


Prob. Show that 2 + 4 + 6 + …… + 2n = n (n+1) by mathematical induction.


Prob. Prove by induction that the sum of the cubes of three consecutive integers is divisible by 9.


Prob. Prove by mathematical induction that n2 + n is an even number for all natural numbers, n ≥ 1.


Prob. Prove by using method of induction-
12 + 22 + 32 +.......+ (3n - 2)2 = (n(6n2 - 3n -1)) /2.


Prob. Prove by mathematical induction -

12 - 22 + 32 -............+ (-1)n+1n2 = ((-1)n+1n(n+1)) /2.
Python Programming ↓ 👆
Java Programming ↓ 👆
JAVA EasyExamNotes.com covered following topics in these notes.
JAVA Programs
Principles of Programming Languages ↓ 👆
Principles of Programming Languages
EasyExamNotes.com covered following topics in these notes.

Practicals:
Previous years solved papers:
A list of Video lectures References:
  1. Sebesta,”Concept of programming Language”, Pearson Edu 
  2. Louden, “Programming Languages: Principles & Practices” , Cengage Learning 
  3. Tucker, “Programming Languages: Principles and paradigms “, Tata McGraw –Hill. 
  4. E Horowitz, "Programming Languages", 2nd Edition, Addison Wesley

    Computer Organization and Architecture ↓ 👆

    Computer Organization and Architecture 

    EasyExamNotes.com covered following topics in these notes.

    1. Structure of desktop computers
    2. Logic gates
    3. Register organization
    4. Bus structure
    5. Addressing modes
    6. Register transfer language
    7. Direct mapping numericals
    8. Register in Assembly Language Programming
    9. Arrays in Assembly Language Programming

    References:

    1. William stalling ,“Computer Architecture and Organization” PHI
    2. Morris Mano , “Computer System Organization ”PHI

    Computer Network ↓ 👆
    Computer Network

    EasyExamNotes.com covered following topics in these notes.
    1. Data Link Layer
    2. Framing
    3. Byte count framing method
    4. Flag bytes with byte stuffing framing method
    5. Flag bits with bit stuffing framing method
    6. Physical layer coding violations framing method
    7. Error control in data link layer
    8. Stop and Wait scheme
    9. Sliding Window Protocol
    10. One bit sliding window protocol
    11. A protocol Using Go-Back-N
    12. Selective repeat protocol
    13. Application layer
    References:
    1. Andrew S. Tanenbaum, David J. Wetherall, “Computer Networks” Pearson Education.
    2. Douglas E Comer, “Internetworking with TCP/IP Principles, Protocols, And Architecture",Pearson Education
    3. KavehPahlavan, Prashant Krishnamurthy, “Networking Fundamentals”, Wiley Publication.
    4. Ying-Dar Lin, Ren-Hung Hwang, Fred Baker, “Computer Networks: An Open Source Approach”, McGraw Hill.