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Mathematical Induction and its Rules


In Mathematical induction, we use the principle by which one can conclude that a statement is true for all positive integers, after proving certain related propositions.

The steps involved are as follows:-

Corresponding to each positive integer n let there be a statement or a proposition P(n).
If (i) P(1) is true, and
   (ii) P(k+1) is true wherever P(k) is true, [ Where k is a positive integer]

Then P(n) is true for all positive integers n.

Note: In some cases, the above properly (1) may start from 2, 3 or 4. In that case check P(2), P(3) or P(4) as the case may be.

Example 1:
Prove by Mathematical Induction:

Solution:
Let P(n) be the given statement.



∴ The result is true for k + 1 if it is true for k.
∴ By Mathematical induction P(n) is true for all natural numbers. 


Example 2:


Solution:
Let P(n) be the given statement.


Using the above formula on RHS of P(k), we get


The result is true for (k + 1) if it is true for k ie P(k + 1) is true if P(k) is true
By mathematical induction, the result is true for all natural numbers.

Example 3:


Solution:
Let the result be represented by


P(1) is true.
Let P(k) be true for k N.

P(k + 1) is true wherever P(k) is true

By Mathematical induction, P(n) is true for all nN.


Example 4:

Prove:
 

Solution:

Hence P(k+1) is true if P(k) is true.

Hence proved.

 

Example 5:


Solution:
Let the given result be P(n), n N


P(k + 1) is true when P(k) is true.
By mathematical induction the result is true for all natural numbers.

 

​Example 6:
Prove by mathematical induction

Solution:

Let P(k) be true for some k N

Simplifying R H S, we get

This is the result P(k + 1).
By mathematical induction P(n) is true for all n N.




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