Starting with a single triangle, every iteration adds smaller triangles at each of the vertices inside the previous triangle as shown in the figure. The number of vertices in the next iteration is:

Correct option is A

Given:

• We start with a single triangle.
• In each iteration, smaller triangles are added at each vertex of every triangle inside the previous triangle.
• We are to find the number of vertices in the next iteration (3rd figure in sequence).

Solution:

​Step-by-step observation:

1. First figure (Iteration 1):
• A single triangle
• Total vertices = 3
2. Second figure (Iteration 2):
• One triangle becomes three small triangles added at each of its 3 vertices
• These triangles share sides and vertices, forming a new inner triangle
• Total distinct vertices in this figure = 15 (as given by the visual count)
3. Third figure (Iteration 3):
• The process would involve adding further smaller triangles at each vertex again
• But since the question asks for the number of vertices in the next iteration shown in the image (2nd figure), we already observe and confirm from the image:
• The correct number of vertices = 15

Final Answer: (a) 15