Any hints to approach this problem?

[turbid cradle]

So when discussing with my classmates, we found out by just pure computation that the statement holds true when n>6 (not including the case of n=0). Then prove by induction. Are there ways where we can find the value of n when this starts to hold true? It does not seem like this method is the intended approach to this problem. As a hypothetical, what if this statement holds true for n>30? Does that mean we should be plugging in values of n until we hit that?

[compact ether]

For this, consider left hand side divided by right hand side.

The quotient decreases as n approaches 3 (why?), and increases thereafter.

Generally you should expect you can intuitively see why it keeps holding from some n, but sometimes it's probably hard to find that n

There are much harder mathematical problems where doing the search is hard.

[turbid cradle]

When you say division, do you mean n! / 3^n >= 1?

[compact ether]

yeah you look at that

and see how incrementing n by 1 chances n!/3^n, so you are looking at

$\frac{\frac{(n+1)!}{3^{n+1}}}{\frac{n!}{3^n}}$

[nova stump]

Element118

[turbid cradle]

So I am comparing how n+1 changes vs n changes, correct?

And I am assuming that when we have the output of the division greater than 1 then we should look at that

[compact ether]

"plug in values" is a technique you are expected to be able to perform when given - most questions that require that wouldn't make you search too much

[compact ether]

you can see how things change carefully, that's a way to improve the intuition for this problem

though I would probably recommend more practice with other problems since there's not much more to look at (besides obvious generalisations of n! > P(n)k^n where P(n) is polynomial in n)

[vapid radish]

test cases

it doesn't take very long for factorials to dominate exponentials