If the mean deviation about the median of the numbers $a, 2 a, . . . ., 50 a$ is $50$ , then $| a |$ equals:-

4

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5

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2

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3

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Solution

The correct option is A $4$
Given series:- $a, 2 a, 3 a, . . . . ., 50 a$
Median $= \frac{25 a + 26 a}{2} = 25.5 a$
Mean deviation about median $= 50 (Given)$
Mean deviation $= \frac{\sum_{i = 1}^{50} | x_{i} - 25.5 a |}{50}$
$∴ \frac{24.5 a + 23.5 a + 22.5 a + . . . . . + 23.5 a + 24.5 a}{50} = 50$
$\Rightarrow a + 3 a + 5 a + . . . . . + 47 a + 49 a = 2500$
$\Rightarrow \frac{25}{2} (2 a + (25 - 1) 2 a) = 2500$
$\Rightarrow 2 a \times 25 = 200$
$\Rightarrow a = \frac{200}{50} = 4$

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