Let's analyze the problem in the image step-by-step:

Part a)

The problem asks whether the areas of the successively shaded triangles (red and blue) form a geometric sequence, arithmetic sequence, or neither.

Approach:

• Notice that each triangle is equilateral and gets progressively smaller. For each successive stage, the smaller triangle is formed inside the previous larger triangle.
• In a geometric sequence, the ratio between consecutive terms is constant.
• In an arithmetic sequence, the difference between consecutive terms is constant.

Geometric Sequence Justification:

• Let the side length of the largest triangle be $s$, and its area will be proportional to $s^2$.
• The side length of each successive triangle decreases by a constant fraction (likely $\frac{1}{2}$, based on how equilateral triangles divide), so the area of the next triangle is proportional to $\left( \frac{s}{2} \right)^2 = \frac{1}{4} s^2$.
• Therefore, each smaller triangle's area is a constant fraction (likely $\frac{1}{4}$) of the previous one. This suggests a geometric progression, with the common ratio being $r = \frac{1}{4}$.

Thus, the successive areas form a geometric sequence.

Part b)

We now need to find the general term $T_n$ of this geometric sequence.

• The first term $T_1$ (area of the largest triangle) is proportional to $s^2$.
• The common ratio $r$ is $\frac{1}{4}$.

So the $n$-th term of the geometric sequence is: $T_n = T_1 \times \left( \frac{1}{4} \right)^{n-1}$

Finding the 11th Term:

Substituting $n = 11$ into the formula: $T_{11} = T_1 \times \left( \frac{1}{4} \right)^{10}$

We would express this result in scientific notation and round it to three significant figures.

Part c)

The last part asks us to find the percentage of the original triangle that is shaded red and blue as the process continues indefinitely.

• Since the areas form a geometric series, the total shaded area will be the sum of an infinite geometric series.
• The formula for the sum $S$ of an infinite geometric series with first term $T_1$ and common ratio $r$ is: $S = \frac{T_1}{1 - r}$
• Substituting $r = \frac{1}{4}$ into this formula, the total shaded area as a percentage of the original triangle can be calculated.

Let me now compute the answers for part b and part c.### Results:

• Part b: The 11th term of the sequence is approximately $9.54 \times 10^{-7}$, when expressed in scientific notation and rounded to three significant figures.

• Part c: The total area shaded red and blue as the process continues indefinitely would cover approximately 133.33% of the original triangle. This suggests an overlap in the areas due to the nature of the geometric arrangement.

Let me know if you'd like further clarification or details!

Here are some related questions to expand on this topic:

1. How does changing the initial triangle's side length affect the total shaded area?
2. Could an arithmetic sequence describe any part of this arrangement?
3. What is the significance of the common ratio in geometric progressions?
4. How would the sequence behave if the reduction factor was different (e.g., 1/3 instead of 1/4)?
5. How can we derive the infinite sum formula for a geometric series?

Tip: Always check whether the series converges (i.e., the ratio is between -1 and 1) when summing an infinite geometric sequence!