I have a example problem that I can't figure out.

We have listed in adjacent collumns the values of 4n and n^2 - 7 for the positive integers n, where 1 <= n <= 8

n |4n| n^2-7

1 | 4 | -6

2 | 8 | -3

3 | 12 | 2

4 | 16 | 9

5 | 20 | 18

6 | 24 | 29

7 | 28 | 42

8 | 32 | 57

From the table, we see that (n^2 - 7) < 4n for n = 1,2,3,4,5 but when n=6,7,8, we have 4n < (n^2-7). These last three observations lead us to conjecture: For all n>=6 4n < (n^2 - 7)

Let S(n) denote the open statement 4n < (n^2 - 7). Then table confirms that s(6) is true and we have our basis step.

In this example, the induction hypothesis is S(k): 4k < (k^2 - 7) where K is positive integer and K >= 6. In order to establish the inductive step, we need to obtain the truth of S(k + 1) frin that if s(k). That is, from 4k < (k^2-7) we must conclude that 4(k+1) < [(k+1)^2 - 7].

Here are the necessary steps

4k < (k^2 - 7) => 4k +4< (k^2 - 7) +4< (K^2 - 7) +(2k + 1)

(because for K >= 6, we find 2k+1 >= 13 > 4), and

4k + 4 < (K^2 - 7) + (2k+1) => 4(k+1) < (K^2 +2k + 1) - 7 = (k+1)^2-7

My question is where in the world do the numbers I bolded come from????