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) : 2k > k  ......  (1)

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

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

⇒  2k+1 > 2k   ......  (2)

Also,  2k > k + 1  for all  k > 1

∴  From equation  (2)  –

2k+1 > 2k > k + 1

⇒  2k+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.

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step 1 .. for n=1

21> 1 true

step 2 ....for n=k

2k > k -->1

step 3 ..to prove for n=k+1

2k+1 = 2k . 2 > k. 2  [using 1]

2k . 2> 2k > k+1  [2k > k+1] for all k>1 ]

= 2k . 2 > k+1

= 2k+1 > k+1   proved

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