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False.
let the integers be n, n+1 and n+2
3n+3 is there sum and we need this to be even for all integers n.
if n is odd, then 3n is odd ( take n=5 3x5=15 odd)
any odd number +3 is even.
if n is even, then 3n is even and an even number plus and 3 is odd so the answer is false
You could just say or prove it is false with a single counter example.
Take the 3 consecutive integers, 2,3,4 their sum is 9 and you are done.
I mentioned the 3n+3 so you can see why it is false for even set of 3 when the first of the 3 consecutive numbers is even.
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