...sorry the first bullet should be 5^n+1 - 4(n+1) -1 = ....
8 divides for all integers n greater than or equal to 1.
Proof:
Basis step:
n = 1
= 0
0/8 = 0
8 divides if n=1
Induction step:
Assume is divisible by 8 (induction hypothesis)
[show 5^(n+1) - 4(n+1) - 1 is divisible by 8]
...now im not sure where to go
A few points.
1. You can use the "Edit" feature rather than posting multiple times, it will keep the thread cleaner.
2. You made a silly error of not distributing the -4 properly, so you should have -5 where you currently have +3.
3. I think the problem can be solved by considering that when n is odd, 4n+1 is congruent to 5 (mod 8), and when n is even, 4n+1 is congruent to 1 (mod 8). That is, we have by induction hypothesis
and so we can find out what is (mod 8), by considering the two cases, n odd or even.
4. The way to get exponents with more than one character in LaTeX is like this (hover mouse over it to see the code): .
Edit: I had a typo; the -4 in red above used to say -1.