Math Help - Induction

1. Induction

By using mathematical induction:
Show (n^4) − 12(n^2) + 25n − 17 > 0 for all integers n ≥ 2.

2. Test it for n=2 to show its positive.

Assume its >0 for n

Set n=n+1 and expand the equation but dont add the parts together. This should give $n^4 - 12n^2 + 25n - 17 + (4n^3 + 6n^2 - 20n + 14)$.

Now do the same process on the part of this equation thats in brackets to prove its also positive, if it is, that means the entire equation above is positive. Hence positive for the inductive case n=2, n and n+1 hence +ve for all $n \geq 2$

It is for n=2;
Assume it is for n;
Let n=n+1 and expand the equation to give $4n^3 + 6n^2 -20n + 14 + (12n^2 + 4n + 4) > 0$ since we assumed $4n^3 + 6n^2 -20n + 14 > 0$ and $12n^2 + 4n + 4$ is clearly greater than 0.

So its positive for the base case n=2, n and n+1, hence +ve for all $n \geq 2$

Sorry i wrote this in a hurry if its unclear let me know.