The problem asks us to solve the following:

Given the sum $A = 1 + 3^2 + 3^4 + 3^6 + \dots + 3^{100}$, calculate the value of the expression $B = 8A - 3^{100}$.

Step 1: Analyze the Series

The sum $A$ is a geometric series with the first term $a_1 = 1$ and the common ratio $r = 3^2 = 9$.

The series can be written as: $A = 1 + 9 + 9^2 + 9^3 + \dots + 9^{50}$

This is a geometric series where:

• The first term $a_1 = 1$
• The common ratio $r = 9$
• The number of terms $n = 51$ (because the powers of 9 start from $9^0$ to $9^{50}$)

Step 2: Calculate the Sum of the Geometric Series

The sum $A$ of the geometric series can be calculated using the formula: $A = \frac{a_1 (r^n - 1)}{r - 1}$ Substituting the values: $A = \frac{1(9^{51} - 1)}{9 - 1} = \frac{9^{51} - 1}{8}$

Step 3: Calculate $B$

Now, the expression for $B$ is given by: $B = 8A - 3^{100}$ Substituting $A$ into the expression for $B$: $B = 8 \left(\frac{9^{51} - 1}{8}\right) - 3^{100}$ Simplifying: $B = 9^{51} - 1 - 3^{100}$

Final Answer

The value of $B$ is: $B = 9^{51} - 3^{100} - 1$

Would you like to dive deeper into any part of the solution, or do you have any further questions?

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Here are 8 related questions to deepen your understanding:

1. What is the general formula for the sum of a geometric series?
2. How can the geometric series formula be derived?
3. What would be the first term if the sum $A$ started with $3^2$ instead of 1?
4. What if the common ratio were 5 instead of 9? How would that affect $A$?
5. How does changing the exponent of 3 in the expression for $B$ affect the final result?
6. Could you simplify $B = 9^{51} - 3^{100} - 1$ further?
7. How can you express $3^{100}$ in terms of powers of 9?
8. How would the series sum change if it extended to $3^{102}$ instead of $3^{100}$?

Tip: When dealing with geometric series, always verify the number of terms and the common ratio, as they are crucial for correctly applying the summation formula.