# Math Help - proof by induction

1. ## proof by induction

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

2. Originally Posted by Rubberduckzilla
Prove by induction that 3^n>n^3 where n>3 please.

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

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

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

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

the last inequality is true because:

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

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

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!!