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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 $35 \frac{7}{17}$. What was the number erased?

  1. $7$
  2. $8$
  3. $9$
  4. None of these
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Sum of all the numbers from 1 to n = n(n+1) / 2 

 Assuming the erased number is x

 After erasing one number remaining numbers = (n-1)

∴ Sum - x = 35 7/17 (n-1)

∴ {n(n+1) / 2} - x = 35 7/17 *(n-1)

As 1 to n all are natural numbers & x is also a natural number between 1 to n

On a series of natural numbers, after erasing one natural number the maximum change in average can be only 0.5

So, post-erasing 

       the average will be in the vicinity of 35 (≈ 35)

       Before erasing the average is in  the vicinity of 35 (≈ 35)

      We know that the average is the middle term of any sequence

      So, the middle term is also in the vicinity of 35 (≈ 35)

∴ The number of terms n has to be in the vicinity of 70 [35*2]

∴ (n-1) also has to be in the vicinity of 70

And

post-erasing

           the series will be 1 to (n-1)

         ∴ Sum / (n-1) = 35 7/17

∴ (n-1) has to be a multiple of 17

    and as (n-1) is in the vicinity of 70

∴ (n-1) = 17 * 4 = 68

∴  n= 69

Now, we'll put the value of n in 1)

   {n(n+1) / 2} - x = 35 7/17 *(n-1)

   69 *70 / 2 - x = [ {(35 *17) + 7} * 68 ] / 17

   2415 - x = 40936 / 17

   x = 2415 - 2408

∴ x = 7

∴ 15 is erased from the sequence or series of  natural numbers from 1 to 69

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