Big O how to calculate the n [closed]

Question

Closed. This question does not meet Stack Overflow guidelines. It is not currently accepting answers.

---

This question does not appear to be about programming within the scope defined in the help center.

Closed 9 years ago.

Improve this question

I am having some problem understanding this solution.

10n^2 + 4n + 2 ≤ 11n^2 for all n ≥ 5,

I could solve this in another way e.g. 10n^2 + 4n + 2 ≤ 16n^2 for all n ≥ 1

But how do we get the n ≥ 5 for the first solution?

Answer 1

It's true by inspection that the inequality doesn't hold for n less than 5.

In [5]: for n in range(1, 6):
   ...:    print 10 * n**2 + 4 * n + 2 <= 11 * n**2
   ...:     
False
False
False
False
True

It makes sense to inspect for small values. Otherwise you can use a little calculus or plot the two functions to see where they intersect.

Answer 2

Simply because 4n + 2 ≤ n^2 for n greater or equal 5. It is true for n=5. If n increases by 1, the left side increases by 4, while the right side increase by a value greater than 5, because (n+1)^2 = n^2 + 2n + 1. So the statement remains true for larger values of n. You can easily check that it is not true for smaller values.