Problem:
What is the greatest number of consecutive integers whose sum is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The sum . This sum has consecutive integers. There is no longer list because for the sum of consecutive integers to be positive, there must be more positive integers than negative integers. Further, if there are more than consecutive integers as part of a list that sums to a positive number, then there must be a positive integer greater than that is not cancelled out by its additive inverse.
Suppose that the consecutive integers are ; their sum then equals . Therefore
so , which implies that . A sequence of consecutive integers with sum equal to indeed exists, as observed in the first solution.
The problems on this page are the property of the MAA's American Mathematics Competitions