1. Analyze the Average of Remaining Numbers:
Let the initial total number of integers written on the board be n. The numbers are 1,2,3,…,n.
After erasing one number, the count of remaining numbers becomes n−1.
The average of the remaining numbers is given as:
Average=35177=1735×17+7=17595+7=17602
The formula for the average is:
Average=n−1Sum of remaining numbers
Since the sum of integers must be an integer value, the denominator (n−1) must be a multiple of the denominator of the fraction in its simplified form, which is 17.
n−1=17,34,51,68,…
n=18,35,52,69,…
2. Determine the Correct Value of n:
The original average of n consecutive natural numbers from 1 to n is:
Original Average=2n+1
When one number is removed from a set, the new average cannot deviate significantly from the original average. Specifically, it remains very close to 2n+1.
Given that the new average is approximately 35.41, let's estimate n:
2n+1≈35.41
n+1≈70.82⟹n≈69.82
From our possible list of values for n (18,35,52,69,…), the closest integer matching this approximation is n=69.
Therefore, n−1=68 (which is a multiple of 17, as 17×4=68).
3. Set Up the Sum Equation:
The sum of all n natural numbers from 1 to 69 is:
Total Sum=2n(n+1)=269×70=69×35=2415
The sum of the remaining 68 numbers is:
Remaining Sum=Average×(n−1)=17602×68
Remaining Sum=602×4=2408
4. Find the Erased Number:
Let the erased number be x.
x=Total Sum−Remaining Sum
x=2415−2408=7
Correct Answer: A) 7
