Prove that 2n > n for all positive integers n by the Principle of Mathematical Induction

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Assume that P(n): 2n> n

If n =1, 21>1. Hence P(1) is true

Let us assume that P(k) is true for any positive integer k,

It means that, i.e.,

2k> k …(1)

We shall now prove that P(k +1) is true whenever P(k) is true.

Now, multiplying both sides of the equation (1) by 2, we get

2. 2k> 2k

Now by using the property,

i.e., 2k+1> 2k = k + k > k + 1

Hence, P(k + 1) is true when P(k) is true.

Therefore, P(n) is true for every positive integer n is proved using the principle of mathematical induction.

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