# Math Help - help me with Inductive Proof #1

1. ## help me with Inductive Proof #1

Here is the given:
$n^2 -7n+12$ is non-negative integer whenever n is an integer with $n \geq 3$

Here is what i come up with:
Hypothesis: $P(k) = k^2 - 7k + 12$
Conclusion: $P(k+3) = (k+3)^2 - 7(k+3) + 12 > 0$

2. Try the base case with n=3.

$3^{2}-7(3)+12=0$

Now, try the induction step:

$(k+1)^{2}-7(k+1)+12$

$=k^{2}+2k+1-7k-7+12$

Rearrange and group:

$=(\underbrace{k^{2}-7k+12}_{\text{this is 0}})+(2k+1-7)$

$\geq 0+2k+1-7=2k-6$

$\geq 2(3)-6=0$

Since n=3 gives 0, then any number larger than 3 will give a result larger than 0 and is non negative.