Induction for inequality

Question

I have to prove by induction this inequality for $n > 10$: $$n-2 < \frac{n^2 - n}{12}$$

I have no idea how to start proving it. I only know that, if n=11 the inequality is true. Now, if $n:= n+1$ (inductive thesis) , what proceeds?

Answer 1

You did the base case, now for the induction step assuming true as hypotesis

$$n-2 < \frac{n^2 - n}{12}$$

we need to show that

$$(n+1)-2 < \frac{(n+1)^2 - (n+1)}{12}$$

using the hypothesis.

Answer 2

Because for all $n\geq11$ we obtain: $$\frac{n^2-n}{12}-(n-2)=\frac{n^2-13n+24}{12}=\frac{(n-11)(n-2)+2}{12}>0.$$