Prove the following by the principle of mathematical induction.

2n?>?n2, where?n?is a positive integer such that?n?> 4.

Solution:

Let the given statement be P(n), i.e.,

P(n) : 2n?>?n2?where?n?> 4

For?n?= 5,

25?= 32 and 52?= 25

?25?> 52

Thus, P(n) is true for?n?= 5.

Let P(n) be true for?n?=?k, i.e.,

2k?>?k2?? (1)

Now, we have to prove that P(k? 1) is true whenever P(k) is true, i.e. we have to prove that 2k? 1?> (k? 1)2.

From equation (1), we obtain

2k?>?k2

On multiplying both sides with 2, we obtain

2 ? 2k?> 2 ??k2

2k? 1?> 2k2

?To prove 2k? 1?> (k? 1)2, we only need to prove that 2k2?> (k? 1)2.

Let us assume 2k2?> (k? 1)2.

? 2k2?>?k2? 2k? 1

??k2?> 2k? 1

??k2?? 2k?? 1 > 0

? (k?? 1)2?? 2 > 0

? (k?? 1)2?> 2, which is true as?k?> 4

Hence, our assumption 2k2?> (k? 1)2?is correct and we have 2k? 1?> (k? 1)2.

Thus, P(n) is true for?n?=?k? 1.

Thus, by the principle of mathematical induction, the given mathematical statement is true for every positive integer?n.

?

IN THIS EXAMPLE IT HAS BEEN WRITTEN THAT ,

To prove 2k? 1?> (k? 1)2, we only need to prove that 2k2?> (k? 1)2.

how?2k? 1?> (k? 1)2?=?2k2?> (k? 1)2

I cant understand how so plz explain that