Mathematical Induction with Inequalities

Question

$ P(n) = n < 3^n - 4 $ for all $ n \ge 2$

Base case: $2 < 3^2 - 4$

$2 < 5$

Inductive step: Assume true for $n = k$, show true for $n = k + 1$

That is, assume $k < 3^{k} - 4$, and show $k + 1 < 3^{k + 1} - 4$

So,

(This is where I might be wrong)

$k + 1 < 3^k + 1 - 4$ (by IH) $\le 3^k + 3^k - 4 = 3^{k + 1} - 4$

Is this a valid proof? I guess I don't understand induction with inequalities very well.

Answer 1

You're very close! Your last equality was incorrect, though. Instead, $$3^k+1-4\le3^k+3^k+3^k-4=3^k\cdot 3-4=3^{k+1}-4.$$

Answer 2

$$k + 1 < 3^k + 1 - 4\lt 3(3^k + 1 - 4)=3^{k+1}-9\le3^{k+1}-4$$