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in Principle of Mathematical Induction by (46.3k points)

For all n ∈ N, by using principle of mathematical induction, then prove that: 2n > n

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Let P(n); 2n > n

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

Let P(m) is true for n = m, i.e.,

P(m) : 2m > m ……(i)

Now, we have to proved that the given statement is also true for n = m + 1 

i.e., P(m + 1) is true

Multiplying by 2 on both sides of equation (i), we have

2 × 2m > 2m

⇒ 2m+1 > 2m = m + m > m + 1

Thus, P(m + 1) is true.

Hence, by the principle of mathematical induction,

P(n) is true for each natural number n ∈ N.

Hence Proved.

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