1997 CEMC Pascal Problems/Problem 10
Contents
Problem
The sum of nine consecutive positive integers is 
. The largest of these integers is
Solution 1
Let 
be the starting number. The full list of numbers is then:
We can now set up an equation involving the sum:
Using the fact that 
for some non-negative integer 
, and the fact that appears 
times, we have:
We can either subtract 
from both sides or factor out a :
or
The problem asks us for the highest number in the list, which is 
.
~anabel.disher
Solution 2
Let be the highest number in the list of consecutive integers. The full list is then:
Using a similar process to solution 1, we get:
~anabel.disher
Solution 3
Let be the middle term in the list. The full list is then:
We can then write an equation for the sum:
We can see that all of the constant terms cancel out, and appears times, so we have:
The largest term in the list is 
.
~anabel.disher
