The sequence shown in the image combines rectangles and triangles. Each rectangle in the sequence is accompanied by two triangles—one on each side.

Solution:

a. How many triangles are there in the shape with 50 rectangles?

• Each rectangle adds 2 triangles to the shape.
• For 50 rectangles: $\text{Number of triangles} = 50 \times 2 = 100.$

b. How many rectangles and triangles are there altogether in that shape?

• Number of rectangles: 50 (as given).
• Total number of triangles: 100 (calculated above).
• Total shapes: $\text{Total shapes} = \text{rectangles} + \text{triangles} = 50 + 100 = 150.$

Final Answer:

• a. 100 triangles
• b. 150 shapes in total

Would you like additional details or explanations? Here are 5 follow-up questions to deepen your understanding:

1. How would the total shapes change if the sequence continued with 100 rectangles?
2. Can you express the total number of triangles as a formula for $n$ rectangles?
3. How does the ratio of rectangles to triangles evolve as the sequence grows?
4. What happens if additional triangles are added between every two rectangles? How would this affect the sequence?
5. Can you identify other sequences in this problem where similar proportional growth occurs?

Tip: When working with sequences, identifying a pattern or formula early simplifies solving similar problems.