Prove 3n < n! by induction using a basis n > 3.

We will use mathematical induction to prove this.

Answer: 3(k + 1) < (k + 1)!

Let's see how we can use mathematical induction to prove this.

Explanation:

For n = 4, evaluate the inequality: 3 × 4 = 12

4! = 4 × 3 × 2 × 1 = 24

So, LHS < RHS

So, it holds for n = 4.

Assume it holds for k (k > 4), which means 3k < k!

Now, we will prove for k + 1.

= 3(k + 1) = 3k + 3

< k! + 3

< (k + 1)!

Thus, this proves 3(k + 1) < (k + 1)!

So, by mathematical induction we proved 3n < n!