I have two questions. Please try to answer both! How many non-similar triangles have angles whose degree measures are distinct positive integers in an arithmetic progression?

In an arithmetic sequence a7 −2a4 = 1, a3 = 0. Find the common difference.

Thank you!

(a)59

(b)

Step-by-step explanation:

Part A

The nth term of an arithmetic sequence,

Let the first angle of the triangle =a

Let the common difference = d.

Since the measures of the angles are in an arithmetic progression

We have:

a + (a + d) + (a + 2d) = 180

3a+3d=180

a+d=60

Since a or d cannot be equal to zero, the minimum possible values of a and d is 1 and the maximum possible values of a and d is 59.

Therefore, there are 59 non-similar triangles.

Part B

In an arithmetic sequence

Common difference,

Therefore: