If the mean deviation about the median of the numbers a,2a,3a,…,50a,is 50, then a equals

If the mean deviation about the median of the numbers $a,2a,3a,\dots ,50a$,is 50, then $\left|a\right|$ equals

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Solution:

The numbers are $a,2a,3a,\dots ,50a$. The median of these numbers is $\frac{25a+26a}{2}=255a.$

$\therefore$ Mean deviation about median

$=\frac{\begin{array}{r}|a-25.5a|+|2a-25.5a|+\dots +|-0.5a|+|0.5a|\\ +\dots +|50a-25.5a|\end{array}}{50}$

$\begin{array}{r}=\frac{|a|}{25}\left(0.5+1.5+2.5+\dots +24.5\right)\\ =\frac{|a|}{25}×\frac{25}{2}\left(0.5+24.5\right)=\frac{25|a|}{2}\end{array}$

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