Key Concepts and Formulas

• Geometric Progression (GP): A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio ($r$). The $n$-th term is $a \cdot r^{n-1}$.
• Sum of a GP: The sum of the first $n$ terms of a GP is given by $S_n = \frac{a(r^n - 1)}{r - 1}$, where $a$ is the first term and $r$ is the common ratio ($r \neq 1$).
• Method of Differences for GP: For a GP, multiplying the series by the common ratio and subtracting the original series from the multiplied series can simplify the sum by canceling out intermediate terms.

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Step-by-Step Solution

Step 1: Identify the Given Series and its Properties Let the given sum be denoted by $S_1$. $S_1 = 2^{10} + 2^9 \cdot 3^1 + 2^8 \cdot 3^2 + \dots + 2^1 \cdot 3^9 + 3^{10}$ To determine the nature of this series, we examine the ratio of consecutive terms: The ratio of the second term to the first term is $\frac{2^9 \cdot 3^1}{2^{10}} = \frac{3}{2}$. The ratio of the third term to the second term is $\frac{2^8 \cdot 3^2}{2^9 \cdot 3^1} = \frac{3}{2}$. Since the ratio between consecutive terms is constant, the series is a Geometric Progression (GP).

We can identify the parameters of this GP:

• The first term ($a$) is $2^{10}$.
• The common ratio ($r$) is $\frac{3}{2}$.
• To find the number of terms ($n$), observe the powers of 2 or 3. The powers of 2 range from 10 down to 0 (in $2^0 \cdot 3^{10}$), which means there are $10 - 0 + 1 = 11$ terms. Alternatively, the powers of 3 range from 0 to 10, also indicating 11 terms. Thus, $n = 11$.

Step 2: Calculate the Sum of the GP using the Formula We use the formula for the sum of the first $n$ terms of a GP: $S_n = \frac{a(r^n - 1)}{r - 1}$. Substituting the values $a = 2^{10}$, $r = \frac{3}{2}$, and $n = 11$: $S_1 = \frac{2^{10} \left( \left(\frac{3}{2}\right)^{11} - 1 \right)}{\frac{3}{2} - 1}$ First, simplify the denominator: $\frac{3}{2} - 1 = \frac{3-2}{2} = \frac{1}{2}$. Next, simplify the term in the parenthesis: $\left(\frac{3}{2}\right)^{11} - 1 = \frac{3^{11}}{2^{11}} - 1 = \frac{3^{11} - 2^{11}}{2^{11}}$. Now, substitute these back into the sum formula: $S_1 = \frac{2^{10} \left( \frac{3^{11} - 2^{11}}{2^{11}} \right)}{\frac{1}{2}}$ To simplify, multiply the numerator by the reciprocal of the denominator (which is 2): $S_1 = 2 \cdot 2^{10} \left( \frac{3^{11} - 2^{11}}{2^{11}} \right)$ $S_1 = 2^{11} \left( \frac{3^{11} - 2^{11}}{2^{11}} \right)$ Cancel out the $2^{11}$ term from the numerator and denominator: $S_1 = 3^{11} - 2^{11}$

Step 3: Relate the Calculated Sum to the Given Equation The problem provides the equation: $2^{10} + 2^9 \cdot 3^1 + 2^8 \cdot 3^2 + \dots + 2 \cdot 3^9 + 3^{10} = S - 2^{11}$. We have calculated the left side of this equation to be $S_1 = 3^{11} - 2^{11}$. Therefore, we can write: $3^{11} - 2^{11} = S - 2^{11}$

Step 4: Solve for S To find the value of $S$, we need to isolate it in the equation. Add $2^{11}$ to both sides of the equation: $3^{11} - 2^{11} + 2^{11} = S - 2^{11} + 2^{11}$ $3^{11} = S$ So, $S = 3^{11}$.

Step 5: Match the Result with the Given Options The calculated value of $S$ is $3^{11}$. Let's compare this with the given options: (A) $\frac{3^{11}}{2} + 2^{10}$ (B) $3^{11} - 2^{12}$ (C) $2 \cdot 3^{11}$ (D) $3^{11}$

Our result $S = 3^{11}$ matches option (D).

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Common Mistakes & Tips

• Identifying the Number of Terms: Carefully count the number of terms. For a series of the form $a^k + a^{k-1}b + \dots + b^k$, there are $k+1$ terms. In this problem, $k=10$, so there are $10+1=11$ terms.
• Algebraic Simplification with Exponents: Be meticulous when simplifying expressions involving exponents, especially when multiplying or dividing terms like $2 \cdot 2^{10}$ or $\frac{2^{10}}{2^{11}}$.
• Formula Application: Ensure the correct GP sum formula is used and that the values for $a$, $r$, and $n$ are correctly substituted.

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Summary

The problem presented a series that was identified as a Geometric Progression. We calculated the sum of this GP using the standard formula for the sum of an $n$-term GP. The calculated sum was then equated to the expression given in the problem statement, which involved $S$. By solving this equation, we found the value of $S$.

The final answer is $\boxed{3^{11}}$ which corresponds to option (D).