PRINCIPLE OF MATHEMATICAL INDUCTION PART 2

Using principle of mathematical induction prove that 2ⁿ > n for all positive integers n.

Let P ( n ): 2ⁿ > n
When n = 1, 2¹ > 1. Hence P ( 1 ) is true.

Assume P ( k ) is true for any positive integer k, then
2^k > k                                 …( 1 )

Multiply both side of equation ( 1 ) by 2, we get

• 2.2^k > 2k
• 2^{k + 1} > 2k
• 2^{k + 1} > k + k
• 2^{k + 1} > k + 1.

Therefore P ( k + 1 ) is true when P ( k ) is true. Hence by the principle of mathematical induction, P ( n ) is true for all n.