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Prove by induction for $n \geq n_0$, $n^3 < 3^n$. What is the value of $n_0$?

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closed as off-topic by Andres Caicedo, T. Bongers, Lord_Farin, azimut, no identity Nov 8 '13 at 9:09

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for n=3, n^3=3^n. Does this help ? –  Claude Leibovici Nov 8 '13 at 7:58
    
@KennedyS It also may be helpful for you to look at$. It will help you in formatting. –  Jeremy Upsal Nov 8 '13 at 8:00

1 Answer 1

up vote 1 down vote accepted

$n^3 < 3^n$ when $n \ge 4$, so $n_0 = 4$. This is our base case since $64 < 81$.

Assume the result to be true for $n=k$, then $k^3 < 3^k\implies 3k^3 < 3^{k+1}$.

We want to show $(k+1)^3 = k^3 + 3k^2 + 3k + 1 \le 3k^3$ for $k\ge4$.

So just show that $2k^3 - 3k^2 - 3k - 1 \ge 0$ for $k\ge4$.

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