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Question:

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?

Explanation:

Let there were n consecutive integers starting with 1 in the original set.

∴ The original average was = $\frac{n\times (n+1)}{2}\times \frac{1}{n}=\frac{n+1}{2}$

∴ Average = 35, if n = 69

∴ Average = 35.5, if n = 70

However, as the new average has 17 in the denominator, we can say that the number of numbers in the new set (n − 1) is 68.

∴ n = 69

∴ Sum of numbers from 1 to 69 = $\frac{69\times 70}{2}$ = 2415

$\because 35\frac{7}{17}=\frac{602}{17}=\frac{2408}{68}$

∴ 68 numbers that remained on the blackboard added up to 2408.

∴ The number that was erased was = 2415 − 2408 = 7

Hence, option (a).