
### Instruction: help me answer this question The difference between any two consecutive interior angles of a polygon is 5°. If the smallest angle is 120°, find the number of the sides of the polygon., think step by step
### Assistant: It's understood from the question that the angles of the polygon will form an A.P. with a common difference d = 5° and first term a = 120°. And, we know that the sum of all angles of a polygon with n sides is 180°(n – 2). Thus, we can say a + (a + d) + (a + 2d) + ... + [a + (n – 2)d] = 180(n – 2) Simplifying the above equation, we get: n * [2a + (n – 2)d] / 2 = 180(n – 2) Since a = 120° and d = 5°, substituting these values in the equation, we get: n * [240 + 5(n – 2)] = 360(n – 2) Further simplification leads us to: n^2 + 47n - 456 = 0 Upon solving the above quadratic equation, we get: n = 9 or n = 16 Therefore, a polygon having 9 or 16 sides will satisfy the condition in the question. , The answer is A polygon having 9 and 16 sides will satisfy the condition in the question.
