# Thread: Prove the following statement by induction: For all n ≥ 4, n³ < 3ⁿ

1. ## Prove the following statement by induction: For all n ≥ 4, n³ < 3ⁿ

I'm probably just being foolish, but I'm having trouble proving it true for n+1, based on the assumption that it's true for n.

2. Originally Posted by feyomi
I'm probably just being foolish, but I'm having trouble proving it true for n+1, based on the assumption that it's true for n.

Hi feyomi,

P(k)

$\displaystyle k^3\ <\ 3^k$

P(k+1)

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

Proof

Examine P(k+1) to see if P(k) being true will cause P(k+1) to be true

$\displaystyle (k+1)^3=k^3+3k^2+3k+1$

$\displaystyle 3^{k+1}=(3)3^k=3^k+3^k+3^k$

Hence, if $\displaystyle k^3\ <\ 3^k$

we ask if $\displaystyle 3k^2+3k+1\ <\ 3^k+3^k$

If $\displaystyle k\ \ge\ 4,\ 3k^2\ <\ 4k^2\ \Rightarrow\ 3k^2\ <\ k^3$

hence we ask if $\displaystyle 3k+1\ <\ 3^k$

$\displaystyle 3k+1\ <\ (k)k^2\ ?$

$\displaystyle 3k\ <\ k^2\ for\ k\ \ge\ 4$

hence $\displaystyle 3k+1\ <\ k^3,\ for\ k\ \ge\ 4$