Note that for a set of consecutive integers, the median is the the average of the first and the last integer

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Let {x, x+1, x+2, x+3, x+4, x+5 } be a set of even number of consecutive integers.

The median M = the average of two middle numbers = [(x + 2) + ( x +3 )] / 2 = (2x + 5)/2

Now the average of the first and last integer = [ x + ( x + 5)]/2 = ( 2x + 5)/2

median = the average of the first and last integer.

Let { x, x +1, x +2, x +3, x +4, x +5, x+6} be a set of odd number of consecutive integers.

The median is the middle integer = x +3

Now the average of the first and last integer = ( x + x +6)/2 = (2x +6)/2 = x+3.

So we proved for even and odd number of consecutive integers. Hence it is true for any set of integers.

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