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

Asked by Alekhya K | 29th Dec, 2013, 02:22: PM

Expert Answer:

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 + n< 2n+1
 
So, from (1), we can say that, we need to show:
2n+1 > 2n> (n+1)2
That is, 2n> n2 + 1 + 2n or n> 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

Answered by  | 29th Dec, 2013, 03:08: PM

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