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+2​n⩽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​