Mathematical Induction


   
 
Illustrative Examples
Example 1:
 
01. If P(n) is the statement n2-n+41 is prime, prove that P(1), P(2) are true but P(41) is not true.
 
Suggested answer:
 
Let P(n) = n2- n + 41 is a prime number
 
P(1) = (1)2- 1 + 41 = 41 is a prime number
 
P(1) is true.
 
P(2) = (2)2 - 2 + 41 = 43 is a prime number
 
P(2) is also true.
 
P(41) = (41)2 - 41 + 41 = (41)2 is a prime number
 
But (41)2 = 41 x 41 = 1681 which is not true.
 
P(41) is false .
 
Remark: Principle of mathematical induction does not hold good in this case. We cannot make a general assertion for any n unless we prove condition (2).
 
Example 2:
 
 
Suggested answer:
 
 
 
 
P(1) is true.
 
 
 
 
 
 
 
 
P(k+1) is also true.
 
Since P(k) is true P(k+1) is true.
 
 
Example 3:
 
 
Suggested answer:
 
 
 
\ P(1) is true.
 
 
 
 
 
 
 
= a multiple of 7+P(k)
 
 
Hence, by P.M.I, P(n) is true for all .
 
Example 4:
 
Prove by P.M.I , n < 2n for all .
 
Suggested answer:
 
 
 
P(1) is true.
 
 
 
 
 
 
P(k+1) is true.
 
Hence, by P.M.I, P(n) is true for all n Î N .
 
  
     
   
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