Problem: Given seven points on a straight line, in the order stated (not necessarily evenly spaced). Let be an arbitrarily selected point on the line and let be the sum of the undirected lengths . Then s is smallest if and only if the point is:
Answer Choices:
A. midway between and
B. midway between and
C. midway between and
D. at
E. at
Solution:
Consider first a single segment : in this case the point can be selected anywhere in the segment since, for any such selection, . Now consider the case of two segments with end-points : the smallest value of is obtained by choosing to coincide with since. for this selection, whereas, for any other selection, .
These two cases are typical, respectively. of odd numbers of segments (that is, even numbers of points) and of even numbers of segments (that is. odd numbers of points). Since the given problem is a -point system, the selection of should be at point .
For a -point system the selection of is anywhere in segment .