Question

If the sum of 4040 consecutive integer numbers is 2020, find out the absolute difference between the first number and the last number.

Answer 1

Let
`x` be the smallest of the consecutive integers summing to 2020. Then the largest integer in the sum is
`x+4039,` so our answer is
`\boxed{4039}.`

For completeness, though, let's solve for the value of
`x` that satisfies the conditions of the problem. Recall that the sum of an arithmetic series with first term
`a_1,` last term
`a_n,` and
`n` terms, is
`(n(a_1+a_n))/(2).` Plugging in our known values gives us the equation
`(4040(2x+4039))/(2)=2020,` which simplifies to
`2x+4039=1.` Thus,
`x=-2019,` so the 4040 consecutive integers are
`-2019,-2018,\dots,-1,0,1,\dots,2018,2019,2020.` Notice how everything except the
`2020` cancels out!