Problem:
The degree measures of the angles of a convex -sided polygon form an increasing arithmetic
sequence with integer values. Find the degree measure of the smallest angle.
Solution:
The problem implies two conditions for , the first term in the sequence, and , the common difference. They are , because all angles must be less than , and , the sum of the measures of the angles of an -gon. Solving for in the second equation yields . Because is an integer, must be even and less than or equal to . Now the average angle in an -gon , and because the angles form an arithmetic sequence, of them must be above . But would force the largest angle to be greater than , which is impossible. Thus the only possible value for is , and , the smallest angle, has measure .