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.