I need to prove that n^4 - 4n^2 is divisible by 3.
The induction hypnosis would be k^4 - 4k^2 is indeed divisible by 3, for k >= 0.
What I don't understand is after we expand out (k+1) -4(k+1)^2, why do we need to subtract (n^4-4n)?
I know that n^4 - 4n = 3t for some integer t (divisible by 3).
The answer is given somewhere in the middle.
Thanks!Hence (n+1)^4 - 4(n+1)2 is divisible by 3 if (n+1)^4 - 4(n+1)^2-(n^4 -4n^2)
is divisible by 3.
At first I thought you were showing me what they did in the solution. Then I started questioning myself "what trick?" and when I worked on the example I actually worked through the example, and found out that we did the same thing.
Yes. Indeed. A nice trick Although I still don't get the reason why they would subtract anything in the first place.