UPSC NDA Math Model Paper 1

The given series a, a + d, a + 2d, ..., a + 2nd is an A.P. with first term a, last term a + 2nd, the common difference d and total number of terms 2n + 1.
Its mean will be [First Term + Last Term]/2 = [a + (a + 2nd)]/2 = a + nd.
Also, the magnitude of the deviations of the terms which are more than mean will be same as of those which are less than mean (the terms of an AP are equidistant from the mean).
∴MD =

|x1−x|+|x2−x|+........+|xn−x|

n

=

n(n+1)|d|

2n+1