Mean deviation about median of numbers

k, 2k, 3k, ..., 1000k

is 500. Find k².

Solution:

There are 1000 terms.

Median:

= average of 500th and 501st terms

= (500k + 501k)/2

= 1001k/2

Mean deviation about median:

MD = (1/1000) Σ|x − median|

For first 1000 natural numbers,

MD about median = 250

Since all terms are multiplied by k:

250k = 500

k = 2

Therefore,

k² = 4

Answer: 4