The diagram below shows an equilateral triangle divided into progressively smaller equilateral triangles.
 


a)	By looking at successive shading (red and blue combined), determine whether the successive shading represents a geometric sequence, an arithmetic sequence, or neither. Justify this mathematically.
b)	Based on your findings from part a) above, state the nth term and use this term to find the 11th term of your series, express your answer to 3 significant figures and in scientific notation. 


c)	If the process of shading (red and blue) the progressively smaller equilateral triangles is continued indefinitely, find the percentage of the original triangle that is now shaded red and blue, express your answer to 4 significant figures. 


### Part (a): Identifying the Sequence Type

Let's analyze the shading process based on the combined red and blue shaded areas across each stage of the division.

1. **First Observation:**
   The entire triangle is shaded (100%).

2. **Second Observation:**
   After dividing the triangle into 4 smaller equilateral triangles, one of the smaller triangles is shaded red and blue, which means \(\frac{1}{4}\) of the original area is shaded in this second stage.

3. **Third Observation:**
   The next stage divides each smaller triangle into 4 even smaller triangles. Thus, the total number of triangles is now 16. One of the triangles is fully shaded red and blue, meaning that \(\frac{1}{16}\) of the original triangle is shaded.

Let’s write the sequence of the shaded areas from each stage:

- First stage: \(1\) (whole triangle shaded).
- Second stage: \(\frac{1}{4}\).
- Third stage: \(\frac{1}{16}\).

Each term of the sequence can be represented as:
\[
1, \frac{1}{4}, \frac{1}{16}, \dots
\]
This is a **geometric sequence** where the first term \(a = 1\) and the common ratio \(r = \frac{1}{4}\).

#### Justification:
- A sequence is geometric if each term is obtained by multiplying the previous term by a constant ratio.
- In this case, multiplying each term by \(\frac{1}{4}\) gives the next term, which confirms that it is a geometric sequence.

### Part (b): Finding the \(n\)th Term

The formula for the \(n\)th term of a geometric sequence is given by:
\[
a_n = a \cdot r^{n-1}
\]
Where:
- \(a\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.

For this sequence:
- \(a = 1\),
- \(r = \frac{1}{4}\).

Thus, the general term for the sequence is:
\[
a_n = 1 \cdot \left(\frac{1}{4}\right)^{n-1} = \left(\frac{1}{4}\right)^{n-1}
\]

Now, to find the 11th term (\(n = 11\)):
\[
a_{11} = \left(\frac{1}{4}\right)^{10} = \frac{1}{4^{10}} = \frac{1}{1,048,576}
\]

Expressing this result to 3 significant figures and in scientific notation:
\[
a_{11} = 9.54 \times 10^{-7}
\]

### Part (c): Percentage of the Original Triangle Shaded (Indefinitely)

Since the process continues indefinitely, we can calculate the total shaded area by summing the infinite geometric series formed by the shaded portions. The sum \(S\) of an infinite geometric series is given by:
\[
S = \frac{a}{1 - r}
\]
Where:
- \(a = 1\) (the first term),
- \(r = \frac{1}{4}\) (the common ratio).

Thus:
\[
S = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3}
\]

The percentage of the original triangle that is shaded red and blue is given by:
\[
\text{Percentage shaded} = S \times 100 = \frac{4}{3} \times 100 = 133.33\%
\]

However, this seems contradictory since a percentage above 100% implies the entire area exceeds the original triangle. It appears there’s a conceptual misunderstanding, likely due to focusing on overlapping regions. Let’s revisit.

The diagram below shows an equilateral triangle divided into progressively smaller equilateral triangles.

a) By looking at successive shading (red and blue combined), determine whether the successive shading represents a geometric sequence, an arithmetic sequence, or neither. Justify this mathematically.
b) Based on your findings from part a) above, state the nth term and use this term to find the 11th term of your series, express your answer to 3 significant figures and in scientific notation.
c) If the process of shading (red and blue) the progressively smaller equilateral triangles is continued indefinitely, find the percentage of the original triangle that is now shaded red and blue, express your answer to 4 significant figures.

Explore the geometric sequence formed by shading progressively smaller equilateral triangles. Learn how to find the nth term, calculate the 11th term to 3 significant figures, and determine the percentage of the triangle shaded indefinitely. Ideal for students in Grades 10-12.

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