1. ## Induction help

Hi I have an induction question. I think I have the thing done properly, I'm just not 100% sure.

The question is: prove that (3n)! <= 27^n (n!)^3
for all positive integers n.

This is what I have so far:

show for 1:
6<27

assume true for k where k is some positive integer
(3k)! <= 27^k (k!)^3

prove true for k+1
(3(k+1))! <= 27^(k+1) (k+1!)^3
(3k + 3)! <= (27^k)(27)((k!)^3)((k+1)^3)
((3k)!)(3k+1)(3k+2)(3k+3) <= (27^k)(27)((k!)^3)((k+1)^3)
THIS IS THE STEP I'M UNSURE ABOUT!!
I take out the original equation.. is this ok?
(3k+1)(3k+2)(3k+3) <= (27)(k+1)^3
27k^3 + 54k^2 + 33k + 6 <= 27k^3 + 81m^2 + 81m + 27

QED?

2. Originally Posted by pikminman
Hi I have an induction question. I think I have the thing done properly, I'm just not 100% sure.

The question is: prove that (3n)! <= 27^n (n!)^3
for all positive integers n.

This is what I have so far:

show for 1:
6<27

assume true for k where k is some positive integer
(3k)! <= 27^k (k!)^3

prove true for k+1
(3(k+1))! <= 27^(k+1) (k+1!)^3
(3k + 3)! <= (27^k)(27)((k!)^3)((k+1)^3)
((3k)!)(3k+1)(3k+2)(3k+3) <= (27^k)(27)((k!)^3)((k+1)^3)
THIS IS THE STEP I'M UNSURE ABOUT!!
I take out the original equation.. is this ok?
(3k+1)(3k+2)(3k+3) <= (27)(k+1)^3
27k^3 + 54k^2 + 33k + 6 <= 27k^3 + 81m^2 + 81m + 27

QED?
Note that $\displaystyle \lim_{n\to\infty}\frac{27(n+1)!^3}{(3n+3)!}\cdot\f rac{(3n)!}{27n!^3}=\frac{1}{27}$. This means that $\displaystyle \frac{27(n!)^3}{(3n)!}\to0$ so clearly at some point $\displaystyle (3n)!>27(n!)^3$. Maybe you want to relook at the question.

3. Originally Posted by Drexel28
Note that $\displaystyle \lim_{n\to\infty}\frac{27(n+1)!^3}{(3n+3)!}\cdot\f rac{(3n)!}{27n!^3}=\frac{1}{27}$. This means that $\displaystyle \frac{27(n!)^3}{(3n)!}\to0$ so clearly at some point $\displaystyle (3n)!>27(n!)^3$. Maybe you want to relook at the question.
I said 27^n not just 27. Maybe I don't understand what you mean.