(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that

[tex]\forall n \in \mathbb{N}: 3^{n} \geq n^{3}[/tex]

3. The attempt at a solution

(1) Show that it is true for n = 1:

[tex]3^{1} \geq 1^{3}[/tex]

(2) Show that if it is true for n = p, then it is true for n = p + 1:

Assume that [tex]3^{p} \geq p^{3}[/tex]

Now,

[tex]3^{p+1} = 3 \cdot 3^{p} = 3^{p} + 3^{p} + 3^{p}[/tex]

[tex](p+1)^{3} = p^{3} + 3p^{2} + 3p + 1[/tex]

Given our assumption, we know that if it could be demonstrated that

[tex]3^{p} + 3^{p} \geq 3p^{2} + 3p + 1[/tex]

then we are done. From here, I'm not sure how to proceed. Should I pull some moves from analysis and argue that certain functions grow faster than others above a certain n? The last inequality is also a stronger criteria, but does not apply to p = 1 or p = 2, since 3^{p} was larger than p^{3}.

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# Homework Help: Even More Induction

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