Question 983960


The sum of the interior angles of a given polygon is the sum of arithmetic progression with the first term  120°  and the common difference of  5°,  according to the condition. 

So,  it is 


{{{(120 + ((n-1)*5)/2)*n}}} degrees. 


From the other side,  it is  180°*(n-2),  according to the formula of sum of interior angles of a polygon.


Thus you have an equation 


{{{(120 + ((n-1)*5)/2)*n}}} = {{{180*(n-2)}}}.


Simplify and solve it step by step:


{{{120n +(n*(n-1)*5)/2}}} = {{{180*n - 360}}},


{{{240n + 5n(n-1)}}} = {{{360n - 720}}},


{{{5n^2 - 5n - 120n +720}}} = {{{0}}},


{{{5n^2 - 125n + 720}}} = {{{0}}},


{{{n^2 - 25n + 144}}} = {{{0}}}.


This quadratic equation has two roots:  n = 9  and  n = 16  (use the quadratic formula or the Vieta's theorem). 


So,  the problem has two solutions:  n = 9  and  n = 16.