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I'm studying for an exam, but am struggling to understand how to prove that something is true if $n$ is sufficiently large. For example, if I'm given $P(n): 2n^3 - 7n^2 \geq 7n -1$, I understand that I need to find an $a$ such that all $n \geq a$ makes $P(x)$ true. I'm just not sure how I would go about doing that.

I tried factoring out the $n$ but I don't see how that would help me. How can I transform an inequality like the above to make it more obvious? Any help is appreciated.

nicons
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    One way is to find some $a$ which you may think works (e.g. because you tried some example $n$) and then proceed by induction. – Wojowu May 09 '20 at 10:34
  • Thanks, wojowu! – nicons May 09 '20 at 10:38
  • "$P(n)$ is true" is equivalent to solving the following inequality for $n$: $2n^3-7n^2-7n+1\geq 0$. While the cubic is somewhat gory, it won't be too difficult to prove that all $3$ real roots are lesser than $5$. Which implies all $n\geq 5$ satisfy, i.e. $a=5$. – AryanSonwatikar May 09 '20 at 10:40
  • But do you need to find such an $a$, or do you only have to prove there exists one? – Bernard May 09 '20 at 11:04

1 Answers1

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For this particular example, one approach without using calculus or induction is:

$2n^3-7n^2 = n^2(2n-7) \gt 4(2n-7) \text{ if } n \gt 2$

and

$4(2n-7) = 8n-28 \gt 7n-1 \text{ if } n \gt 27$

so if $n \gt 27$ we have

$2n^3 - 7n^2 \gt 8n - 28 \gt 7n - 1$

gandalf61
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