is divisible by 8.

We only need to show that the difference f(n+1)-f(n) is divisible by 8. 
For if that difference is divisible by 8 then the assumption that f(n) 
is divisible by 8 will show that f(n+1) is divisible by 8 because the 
sum of the difference and f(n) will be f(n+1) and the sum of two 
multiples of 8 is a multiple of 8. 



}






But we must now show that 

 is divisible by 4 
 is divisible by 4

As above, we only need to show that the difference g(n+1)-g(n) is divisible by
4. For if that is divisible by 4 then the assumption that g(n) is divisible
by 4 will show that g(n+1) is divisible by 4 because the sum of two
multiples of 4 is a multiple of 4.










The expression in parentheses is even, because
all products and powers of odd numbers are odd numbers,
and the difference of two odd numbers is an even number.
Therefore  is a multiple of 4.

Therefore  is a multiple of 8.

Edwin