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 .