Prove by induction that n^3 - n is divisible by 6.
Would I get full marks if I have done this:
let
un=n3−n and assume it is divisible by 6.
Therefor,
un+1=n3+3n2+2nThus,
un+1−un=3n2+3nAs we assumed
un was divisible by 6 and clearly
3n2+3n is divisible by 6 (
as 3 is a factor of 6) then this proves that
un+1 is divisible by 6.
For n=1 we have 1-1=0, which is divisible by 6.
I am not 100% sure on the part where I use the factor of 3 to show that it is divisible by 6, I couldn't get a 6 or a multiple of 6.
I would highly appreciate it if people could give me the general rule about if it is okay to use factors of the number we are trying to show it is divisible by and not the number itself or a multiple.
I am also unsure about the end where I say 0 is divisible by 6, is 0 a positive integer ?
Edit: I would highly appreciate some general advice about what to do after getting U(n) and U(n+1), I struggle to work out whether to add them, subtract them, add one to a multiple of another (which multiples can we use ?), can we multiply both U(n) and U(n+1) but different multiples......
Thnx