# Technical Document Extraction: Geometric Sequence Problem
## Problem Statement
**Title:** The sum of the first $n$ terms in the infinite geometric sequence $\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$ is $\frac{255}{512}$. Find $n$.
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## Textual Analysis
### Key Components
1. **Mathematical Expressions:**
- Sum formula: $\frac{a(1 - r^n)}{1 - r}$ (where $a$ = first term, $r$ = common ratio)
- Target sum: $\frac{255}{512}$
- Sequence terms: $\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$
2. **Solution Steps:**
- **Step 1:** Identify $a = 1$ (first term) and $r = \frac{1}{2}$ (common ratio).
- **Step 2:** Set up equation:
$\frac{1 - (\frac{1}{2})^n}{1 - \frac{1}{2}} = \frac{255}{512}$
- **Step 3:** Simplify:
$1 - (\frac{1}{2})^n = \frac{255}{256}$
- **Step 4:** Solve for $n$:
$(\frac{1}{2})^n = \frac{1}{256} \Rightarrow n = 8$
3. **Contradiction in Final Answer:**
- The image explicitly states:
**"So we get $n=7$."**
This conflicts with the mathematical derivation ($n=8$).
4. **Scoring Annotations:**
- Multiple `[score = X.X]` tags indicate partial credit for steps:
- `[score = 0.10]` for setting up the sum formula.
- `[score = 0.15]` for recognizing the fraction $\frac{255}{512}$.
- `[score = 0.30]` for the final answer (incorrectly marked as $n=7$).
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## Spatial Grounding & Trends
- **Legend:** No explicit legend present.
- **Text Placement:**
- Problem statement: Top-left corner.
- Solution steps: Centered with LaTeX-style math notation.
- Scoring annotations: Right-aligned in brackets.
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## Component Isolation
### Header
- **Text:** "The sum of the first $n$ terms in the infinite geometric sequence $\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$ is $\frac{255}{512}$. Find $n$."
### Main Chart
- **Mathematical Derivation:**
- Sequence terms: $\frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$ (geometric progression with $r = \frac{1}{2}$).
- Sum formula application:
$\frac{1 - (\frac{1}{2})^n}{1 - \frac{1}{2}} = \frac{255}{512}$.
### Footer
- **Final Answer:**
- Incorrectly stated as $n=7$ despite derivation showing $n=8$.
- Scoring annotation: `[score = 0.30]`.
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## Data Table Reconstruction
| Step | Description | Score |
|------|-------------|-------|
| 1 | Sum formula: $\frac{a(1 - r^n)}{1 - r}$ | 0.10 |
| 2 | Set sum equal to $\frac{255}{512}$ | 0.15 |
| 3 | Simplify to $1 - (\frac{1}{2})^n = \frac{255}{256}$ | 0.15 |
| 4 | Solve for $n$: $(\frac{1}{2})^n = \frac{1}{256} \Rightarrow n=8$ | 0.30 (incorrectly marked as $n=7$) |
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## Trend Verification
- **Sequence Behavior:**
- Terms decrease exponentially ($r = \frac{1}{2}$).
- Sum approaches 1 as $n \to \infty$.
- **Derived Value:**
- $n=8$ satisfies $\frac{255}{512} = 1 - \frac{1}{256}$.
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## Conclusion
The problem contains a critical error: the final answer $n=7$ contradicts the mathematical derivation ($n=8$). The scoring annotations suggest partial credit for intermediate steps, but the conclusion is inconsistent with the calculations.
