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Exam-Style Question on Sequences Exponential ModelsA mathematics exam-style question with a worked solution that can be revealed gradually |
Question id: 121. This question is similar to one that appeared on a IGCSE Extended paper in 2014. The use of a calculator is allowed.
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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