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To solve this problem, let's analyze the given information and determine the volume of each pyramid in the sequence.

### Step 1: Understanding the Volume Formula
The volume \( V \) of a pyramid with a square base of side length \( s \) and height \( h \) is given by:
\[
V = \frac{1}{3} \times s^2 \times h
\]

### Step 2: Initial Pyramid Information
For the first pyramid:
- **Volume**: 200 cubic units
- **Height**: \( h = \frac{3}{4} s \), where \( s \) is the side length of the base.

Using the volume formula for the first pyramid, we substitute \( h = \frac{3}{4} s \):
\[
200 = \frac{1}{3} \times s^2 \times \frac{3}{4} s
\]
\[
200 = \frac{1}{4} s^3
\]
Solving for \( s \):
\[
s^3 = 800 \Rightarrow s = \sqrt[3]{800} = 8.9282 \, (\text{approx.})
\]

### Step 3: Volume Reduction in Sequence
Each subsequent pyramid has a base area that is half of the previous pyramid’s base area. Since the area of a square is \( s^2 \), this means each new side length \( s \) will reduce by a factor of \( \frac{1}{\sqrt{2}} \).

Thus, each pyramid in the sequence has a volume that is half of the previous pyramid’s volume.

### Step 4: Volume Sequence and Total Volume
Since each pyramid’s volume is half of the one before it, this forms a geometric sequence where:
- **First term \( V_1 = 200 \) cubic units**
- **Common ratio \( r = \frac{1}{2} \)**

The sum of the first \( n \) terms of a geometric sequence is:
\[
S_n = V_1 \frac{1 - r^n}{1 - r}
\]

For the first 5 pyramids:
\[
S_5 = 200 \times \frac{1 - \left( \frac{1}{2} \right)^5}{1 - \frac{1}{2}}
\]
\[
S_5 = 200 \times \frac{1 - \frac{1}{32}}{\frac{1}{2}}
\]
\[
S_5 = 200 \times \frac{\frac{31}{32}}{\frac{1}{2}}
\]
\[
S_5 = 200 \times \frac{31}{16} = 387.5
\]

### Final Answer
The total volume of the first 5 pyramids is **387.5 cubic units**.

Would you like more details on any step or have additional questions?

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Here are some related questions:
1. How would the total volume change if the base reduction factor was different?
2. What if the height was a different fraction of the base side length?
3. How does the volume formula change if the base is not square?
4. What is the impact of altering the common ratio in a geometric sequence?
5. How can we calculate the side length from a given volume?

**Tip**: In geometric sequences, understanding the common ratio’s impact helps in quickly determining patterns for sums and growth rates.

A sequence of pyramids is constructed such that each pyramid has a square base and a height that is 3/4 of the length of a side of its base. The volume of the first pyramid is 200 cubic units. Each subsequent pyramid in the sequence has a base that is 1/2 the area of the base of the previous pyramid. What is the total volume of the first 5 pyramids in the sequence?

Explore the process of finding the total volume of a sequence of pyramids with square bases, where each pyramid's base area is halved in each step. This solution covers concepts of geometric sequences and pyramid volume calculations, ideal for students in Grades 9-12.

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