The exterior angles of a k-sided polygon form an arithmetic sequence. The smallest and largest interior angles of the polygon are 136 degrees and 145 degrees. What is k?
The exterior angles of the polygon form an arithmetic sequence.
Then the interior angles of the polygon should also form an arithmetic sequence.
The first term of that arithmetic sequence is 136 and the last term is 145, and there are k terms.
Suppose that the common difference is d degrees. Then \(136 + (k - 1) d = 145\).
That means \((k - 1)d = 9\). Then k - 1 is a factor of 9. Therefore, k can only be 2, 4, or 10.
Obviously, 2-sided polygons are not a thing, and the interior angles are too large for a quadrilateral.
Then k = 10.
You can try to figure out how to present this in a rigorous way.