Question 37

A set of consecutive positive integers beginning with 1 is written on the blackboard. A student came along and erased one number. The average of the remaining numbers is $$\frac{602}{17}$$. What was the number erased?

Solution

Since the number starts from 1 if there are n numbers then initial average = $$\dfrac{n+1}{2}$$.

Average of N natural number can be either an integer {ab} or {ab.50} type. For example average of first 10 number = 5.5 whereas the average of first 11 natural numbers is 6. 
Even if we erased the largest number change in average will be always less than 0.5. 
Here we are given the average is 602/17 or 35$$\frac{7}{17}$$ Hence we can say that average must have been 35.5 or 35 before. 
Case 1: If the average was 35.5 before the erasing process. 
We know that average of 1st N natural number = $$\dfrac{N+1}{2}$$
35.5 = $$\dfrac{N+1}{2}$$
N = 70. 
Sum of these 70 numbers = 70*71/2 = 35*71 = 2485.
Sum of the 69 numbers which we are left with after removing a number = (602/17)*69 = 2443.41. Which is not possible as the sum of natural numbers will always be an integer. Hence, we can say that case is not possible. 

Case 1: If the average was 35 before the erasing process. 
We know that average of 1st N natural number = $$\dfrac{N+1}{2}$$
35 = $$\dfrac{N+1}{2}$$
N = 69. 
Sum of these 69 numbers = 69*70/2 = 35*69 = 2415.
Sum of the 68 numbers which we are left with after removing a number = (602/17)*68 = 2408. 
Hence, we can say that the erased number = 2415 - 2408 = 7. 


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