Question

Prove that 2 1 n 2 n 2
n n 1
> + ∀ >
−
, using the inequalities of Unit 6.

Answer 1

Well, in general case:

$$2n^2>n^2-n-1.$$ $$2n^2-n^2+n+1>0.$$ $$n^2+n+1>0.$$

Now we have a quadratic inequality. The quadratic equation

$$n^2+n+1=0\\$$

has the following discriminant:

$$D=1^2-4*1*1=-3\\$$

$D<0$, therefore quadratic equation has no real roots (only complex ones). As far as the coefficient before the second power is positive ($a=1$), parabola, which represents equation, is oriented upward, thereby confirming inequality $\, n^2+n+1>0$ for $\forall$ $n\in\real$.

In particular cases of integers:

1. For $n\geq0$ inequality $n^2+n+1>0$ is obviously true.
2. For $n<0$. Let $k=-n>0$. Then we can rewrite inequality:

$k^2-k+1>0\\$

$k^2-k=k(k-1)>0$ for any $k>1.\\$

Thus, inequality $n^2+n+1>0$ is true for $\forall$ $n\in\mathbb{Z}$ .