Prove by induction that 3^n>n^3 where n>3 please.

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- May 27th 2009, 02:39 PMRubberduckzillaproof by induction
Prove by induction that 3^n>n^3 where n>3 please.

- May 27th 2009, 03:23 PMapcalculus

Verify n = 4 : $\displaystyle 3^4 > 4^3 $ sure enough, true.

Assume true for n = k - 1 : $\displaystyle 3^{k-1} > (k-1)^3$

Prove for n = k: $\displaystyle 3^k > k^3$

Left Side = $\displaystyle 3^k = 3 3^{k-1} > 3 (k-1)^3 > k^3$

the last inequality is true because:

$\displaystyle \frac{1}{3} < (\frac{k-1}{k})^3$

$\displaystyle \frac{1}{3} < (1 - \frac{1}{k})^3$

When k = 4, we have 1/3 < 9/16

When k = 5, we have 1/3 < 16/25

as k goes to infinity: we have (1/3) < 1

Good luck!!