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Arbitary arrange ment to get the sum

Source: BdMO National 2016

February 26, 2019
number theoryalgebraBdmocontests

Problem Statement

BdMO National 2016 Higher Secondary
Problem 4: Consider the set of integers {1,2,.........,100} \left \{ 1, 2, ......... , 100 \right \} . Let {x1,x2,.........,x100} \left \{ x_1, x_2, ......... , x_{100} \right \} be some arbitrary arrangement of the integers {1,2,.........,100} \left \{ 1, 2, ......... , 100 \right \}, where all of the xix_i are different. Find the smallest possible value of the sum,
S=∣x2−x1∣+∣x3−x2∣+................+∣x100−x99∣+∣x1−x100∣S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | .
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