Let S be the set of all values of a_1 for which the mean deviation about the mean of 100 c…

Question

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of 100 consecutive positive integers $a_1, a_2, a_3, \ldots ., a_{100}$ is 25 . Then $S$ is :

Accepted Answer

Let $a_1$ be any natural number; $a_1, a_1+1, \ldots, a_1+99$ are the values.

$$\bar{x} = a_1+\frac{99\times100}{2\times100} = a_1+\frac{99}{2}$$

$$\text{M.D.} = \frac{1}{100}\left[\frac{99}{2}+\frac{97}{2}+\cdots+\frac{1}{2}\right]\times2 = \frac{1+3+\cdots+99}{100} = \frac{50^2}{100}=25$$

This holds for every natural number $a_1$, so $S = \mathbb{N}$.

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