Question

Prove that $n^2-7n+12$is nonnegative whenever n is an integer with $n\ge 3$

Short answer

$n\ge 3$is a positive integer, then $n^2-7n+12$

Step-by-step solution

Step: 1

If n=3,

$\begin{array}{r}{3}^2-7.3+12\\ 21-21=0\end{array}$

it is true for n=3.

Step: 2

Let P(k) be true.

$k^2-7k+12$

We need to prove that P(k+1) is true.

Step: 3

$\begin{array}{r}(k+1{)}^2-7(k+1)+12\\ =k^2-5k+6\\ =(k-2)(k-3)\\ \ge 0\end{array}$

It is true for P (k+1) is true