# Technical Document Extraction: Mathematical Problem and Solution ## 1. Document Overview This image contains a mathematical word problem followed by a step-by-step derivation and solution. The text includes LaTeX-style mathematical notation and "score" metadata for each step, suggesting it may be from a machine learning evaluation or an automated tutoring system. ## 2. Header: Problem Statement **Text:** "The sum of the first $n$ terms in the infinite geometric sequence $\left\{\frac{1}{4},\frac{1}{8},\frac{1}{16},\dots \right\}$ is $\frac{255}{512}$. Find $n$." --- ## 3. Main Content: Step-by-Step Derivation The following is a transcription of the solution steps provided in the image, including the associated scores. | Step | Mathematical Derivation / Reasoning | Score | | :--- | :--- | :--- | | 1 | Since the common ratio in each term is $\frac{1}{2}$, the sum of the first $n$ terms of this geometric sequence is $\frac{a_1(1-r^n)}{1-r}$, where $a_1$ is the first term and $r$ is the common ratio. | 0.09 | | 2 | So, the sum of the first $n$ terms of this geometric sequence is $\frac{\frac{1}{4}\left(1-\left(\frac{1}{2}\right)^n\right)}{1-\frac{1}{2}}$. | 0.09 | | 3 | So, the sum of the first $n$ terms of this geometric sequence is $\frac{\frac{1}{4}\left(1-\left(\frac{1}{2}\right)^n\right)}{\frac{1}{2}}$. | 0.11 | | 4 | Right. And we want $s_n = \frac{255}{512}$. So we have $\frac{1-\left(\frac{1}{2}\right)^n}{2} = \frac{255}{512}$. | 0.14 | | 5 | We can multiply both sides by 2. | 0.10 | | 6 | We have $1-\left(\frac{1}{2}\right)^n = \frac{255}{256}$. | 0.14 | | 7 | So $\left(\frac{1}{2}\right)^n = 1 - \frac{255}{256}$. | 0.16 | | 8 | So, $\left(\frac{1}{2}\right)^n = \frac{1}{256}$. | 0.30 | | 9 | And then, $2^{-n} = 2^{-8}$. | 0.80 | | 10 | So $-n = -8$. | 0.78 | | 11 | Right. And since the bases are the same, the exponents must be equal. | 0.83 | --- ## 4. Footer: Final Answer **Text:** "#### 8 [score = 0.69]" --- ## 5. Technical Summary of Logic The document describes the following mathematical process: 1. **Identification:** Identifies the sequence as geometric with $a_1 = 1/4$ and $r = 1/2$. 2. **Formula Application:** Uses the geometric series sum formula $S_n = \frac{a_1(1-r^n)}{1-r}$. 3. **Substitution:** Substitutes the known values into the formula and sets the sum equal to $255/512$. 4. **Simplification:** * Simplifies the denominator $(1 - 1/2)$ to $1/2$. * Simplifies the fraction $(1/4) / (1/2)$ to $1/2$. * Solves the resulting linear equation for the term $(1/2)^n$. 5. **Logarithmic/Exponential Solving:** Converts both sides of the equation to base 2 ($2^{-n} = 2^{-8}$) to isolate the variable $n$. 6. **Conclusion:** Determines that $n = 8$.

# Technical Document: Geometric Sequence Sum Analysis ## Problem Statement Find the sum of the first $n$ terms in the infinite geometric sequence $\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$ and determine $n$ when the sum equals $\frac{255}{512}$. --- ### Key Components 1. **Sequence Definition** - Terms: $\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$ - First term ($a_1$): $\frac{1}{4}$ - Common ratio ($r$): $\frac{1}{2}$ (derived from $\frac{\text{Term}_{k+1}}{\text{Term}_k}$) 2. **Sum Formula** The sum of the first $n$ terms of a geometric sequence is: $$ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} $$ 3. **Target Sum** $$ S_n = \frac{255}{512} $$ --- ### Step-by-Step Solution 1. **Substitute Known Values** $$ S_n = \frac{1}{4} \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{1 - \frac{1}{2}} = \frac{1}{4} \cdot \frac{1 - \left(\frac{1}{2}\right)^n}{\frac{1}{2}} = \frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^n\right) $$ 2. **Set Equation Equal to Target Sum** $$ \frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^n\right) = \frac{255}{512} $$ 3. **Solve for $n$** - Multiply both sides by 2: $$ 1 - \left(\frac{1}{2}\right)^n = \frac{255}{256} $$ - Rearrange: $$ \left(\frac{1}{2}\right)^n = \frac{1}{256} $$ - Express $\frac{1}{256}$ as a power of $\frac{1}{2}$: $$ \left(\frac{1}{2}\right)^n = \left(\frac{1}{2}\right)^8 \implies n = 8 $$ --- ### Verification - **Infinite Series Limit**: The infinite sum converges to $\frac{a_1}{1 - r} = \frac{1/4}{1 - 1/2} = \frac{1}{2}$. The target sum $\frac{255}{512} \approx 0.498$ is very close to $\frac{1}{2}$, confirming $n=8$ is reasonable. - **Term-by-Term Calculation**: Summing the first 8 terms: $$ \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots + \frac{1}{256} = \frac{255}{512} $$ --- ### Final Answer - **Sum of First $n$ Terms**: $$ S_n = \frac{1}{2} \left(1 - \left(\frac{1}{2}\right)^n\right) $$ - **Value of $n$**: $$ \boxed{8}