Question 10 Prove by PMI,2^n > n^2,nis greater than or equal to 5 n belong toN In Mathematics Asked by: shiv ,

Assuming the statement to be true for n, we need to prove it for (n+1).

That is, we need to show that 2ⁿ⁺¹ > (n+1)²             ... (1)

 

Now, by induction hyothesis, we have:

n² < 2ⁿ

Multiplying by 2, we get:

2n² = n² + n² < 2ⁿ⁺¹

 

So, from (1), we can say that, we need to show:

2ⁿ⁺¹ > 2n² > (n+1)²

That is, 2n² > n² + 1 + 2n or n² > 1 + 2n

 

Thus, now we need to show that 1 + 2n < n², which can easily be proved by considering the inequalities 

0 < n < n+1 and 0 < 2 < n - 1