PROVE THAT 2 ⁿ > n FOR ALL POSITIVE INTEGERS n.

To prove :  2ⁿ > n  for all positive integers  n.

Consider P(n) : 2ⁿ > n

We will prove the result by the principle of mathematical induction.

For  n = 1 ,  P (1) :  2¹ > 1 which is true.

Suppose the result is true for  n = k.

i.e.  P(k) : 2^k > k  ......  (1)

Now, we will prove that result also holds for  k + 1  if it is true for k.

Since,  2^k+1 = 2^k × 2  >  k × 2  [ Using (1) 2^k > k ]

⇒  2^k+1 > 2k   ......  (2)

Also,  2k > k + 1  for all  k > 1

∴  From equation  (2)  –

 2^k+1 > 2k > k + 1

⇒  2^k+1 > k + 1

⇒  Result is also true for  k + 1  if it is true for k .

Hence,  by principle of mathematical induction, result is true for all positive integers  n.