Problem:
A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is 35177. What number was erased?
Answer Choices:
A. 6
B. 7
C. 8
D. 9
E. can not be determined
Solution:
Let n be the last number on the board. Now the largest average possible is attained if 1 is erased; the average is then n−12+3+⋯+n=n−12(n+1)n−1=2n+2. The smallest average possible is attained when n is erased; the average is then 2(n−1)n(n−1)=2n. Thus
2n⩽35177⩽2n+2n⩽701714⩽n+2681714⩽n⩽701714
Hence n=69 or 70. Since 35177 is the average of (n−1) integers, (35177)(n−1) must be an integer and n is 69. If x is the number erased, then
68269(70)−x=35177 So 69⋅35−x=(35177)68=35⋅68+2835−x=28x=7