## Exam-Style Question on Sequences Exponential Models## A 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 | 2^{0} |
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 | 2^{0} |
2^{2} |
2^{4} |

(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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