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Question id: 121. This question is similar to one that appeared in a IGCSE Extended paper in 2014. The use of a calculator is allowed.

Sequences Exponential Models

Triangle Fractal

The diagrams above show a growing fractal of triangles. The sides of the largest equilateral triangle in each diagram are of length 1 metre.

In the second diagram there are four triangles each with sides of length \(\frac{1}{2}\) metre.

In the third diagram there are 16 triangles each with sides of length \(\frac{1}{4}\) metre.

(a) Complete this table for more diagrams.

Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Diagram 6   Diagram \(n\)
Length of Side 1 \(\frac{1}{2}\) \(\frac{1}{4}\)  
Power of 2 20 2-1 2-2  

(b) Complete this table for the number of the smallest triangles in diagrams 4, 5 and 6.

Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Diagram 6   Diagram \(n\)
Number of smallest triangles 1 4 16
Power of 2 20 22 24

(c) Calculate the number of the smallest triangles in the diagram where the smallest triangles have sides of length \(\frac{1}{256}\) metre.

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