Is 2000 the sum of four consecutive integers?

Answer 1

No.

Let the four numbers be (n-1), n, (n+1), (n+2).

Their sum is 4n + 2 = 2(2n + 1)

If 2000 is the sum of four consecutive integers, then:

2(2n+1) = 2000

⇒ 2n + 1 = 1000

⇒ 2n = 999

but 999 is odd, not even and so n cannot be an integer; therefore 2000 is not the sum of four consecutive integers.