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Let's solve the problem step by step.

### Problem Recap
The problem asks us to:
1. **Calculate the sum \( A = 1 + 3^2 + 3^3 + \cdots + 3^{100} \)**.
2. **Determine the value of the expression \( B = 8A - 3^{101} \)**.

### Step 1: Analyze the Sum \( A \)

The sum \( A \) can be rewritten as:

\[
A = 1 + \sum_{k=2}^{100} 3^k
\]

The sum from \( k=2 \) to \( k=100 \) is a geometric series where:
- The first term \( a_1 = 3^2 \).
- The common ratio \( r = 3 \).
- The number of terms \( n = 99 \).

The sum of a geometric series is given by:

\[
S_n = a_1 \frac{r^n - 1}{r - 1}
\]

Here:
\[
S_{99} = 3^2 \cdot \frac{3^{99} - 1}{3 - 1} = 9 \cdot \frac{3^{99} - 1}{2}
\]

Thus, the full sum \( A \) is:

\[
A = 1 + \frac{9(3^{99} - 1)}{2} 
\]

### Step 2: Calculate \( B = 8A - 3^{101} \)

We now substitute the value of \( A \) into the expression for \( B \):

\[
B = 8 \left( 1 + \frac{9(3^{99} - 1)}{2} \right) - 3^{101}
\]

Expanding this:

\[
B = 8 + 36(3^{99} - 1) - 3^{101}
\]

Simplifying further:

\[
B = 8 + 36 \cdot 3^{99} - 36 - 3^{101}
\]

\[
B = -28 + 36 \cdot 3^{99} - 3^{101}
\]

Notice that \( 36 \cdot 3^{99} = 4 \cdot 3^{101} \), so we can factor:

\[
B = -28 + 3^{101}(4 - 1)
\]

\[
B = -28 + 3 \cdot 3^{101}
\]

\[
B = -28 + 3^{101} \cdot 3
\]

Thus, the final simplified form is:

\[
B = 2 \cdot 3^{101} - 28
\]

### Final Answer:
\[
B = 2 \cdot 3^{101} - 28
\]

This is the value of the expression \( B \).

Do you want more details on any step, or do you have any questions?

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Here are 8 related questions that expand on this topic:
1. What is a geometric series, and how is its sum formula derived?
2. How does the common ratio affect the sum of a geometric series?
3. What happens to the sum if the common ratio is less than 1?
4. How can you approximate the sum of a large geometric series?
5. What are some real-world applications of geometric series?
6. How does the value of \( B \) change if the exponent in the expression is altered?
7. Can the expression \( B \) be simplified further or represented differently?
8. What other types of series are there besides geometric series?

**Tip:** In problems involving series, always check if you can factor out common terms or simplify expressions before performing lengthy calculations.

Explore the solution to a mathematical problem involving the sum of a geometric series and the calculation of an expression. Learn how to calculate complex series and manipulate exponential expressions step-by-step. Suitable for advanced high school students and those preparing for competitive exams.

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