Final answer:
To prove n! > 3^n for n > 7 using induction, we show it holds for the base case and then prove it holds for the next case.
Step-by-step explanation:
To prove that n! > 3n for n > 7 using induction, we need to show that it holds for the base case (n = 8) and then assume it holds for some arbitrary value k and prove it holds for k+1.
For the base case, we have 8! = 8x7x6x5x4x3x2x1 = 40,320, and 38 = 6,561. Since 40,320 > 6,561, the inequality holds for n = 8.
Assuming it holds for some k, we have k! > 3k. Multiplying both sides by (k+1), we get (k+1)! > 3k+1. Since (k+1)! = (k+1) * k!, and we assumed k! > 3k, we can conclude that (k+1)! > 3k+1. Therefore, by induction, n! > 3n for n > 7.