Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 35, pg 330].
a) Use mathematical induction to prove that $n^2$ - 1 is divisible by 8 whenever n is an odd positive integer.
I defined predicate P(n) as $n^2$- 1 is divisible by 8
My basis step was showing P(1), as 1 is the first odd positive integer
P(1) - 8 | $1^2$ - 1 or 8 | 0 which is true
My inductive hypothesis is to assume P(k) for some odd positive integer k and use that to show P(k+2)
So inductive hypothesis - P(k) - $k^2$ - 1 = 8c where s is some integer
Now the next odd positive integer is at k+2 so we have
8| (k + 2)$^2$ - 1
8 | k$^2$ - 1 + 4k + 4
I was able to recognize the k$^2$ - 1 segment from my inductive step so I made an immediate substitution for that, now we have
8 | 8c + 4k + 4.
I recognized that the 8c portion was divisible by 8 but what mathematical steps can you go to show that 4k + 4 is also divisible by 8?