Let n be an integer. Then any number of this form n(n+1) (n+2) is divisible by 6?

So what is your question?

The function is zero at n=0, so that is not true in all cases. At n=1 it becomes 1x2x3=6 therefore true in all positive non-zero cases. Not true in all negative cases. Here is the plot:
www.wolframalpha.com/input/?i=plot+y%3Dx(x%2B1)(x%2B2)(x%2B3)