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Home » Let the cost of a notebook be \( n \) and a pen be \( p \). - Dachbleche24

Let the cost of a notebook be \( n \) and a pen be \( p \). - Dachbleche24

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 For $(1,25)$: $k - y = 1$, $k + y = 25$ → $k = 13$, $y = 12$ → $x = 26$.

📌 For $(5,5)$: $k - y = 5$, $k + y = 5$ → $k = 5$, $y = 0$ → $x = 10$.

📌 Negative pairs yield symmetric solutions. Total lattice points: $(\pm26, \pm12)$ and $(\pm10, 0)$, totaling $\boxed{6}$.Question: A mechanical engineer is designing a gear system where the torque output $ T(n) $ at segment $ n $ is defined recursively by $ T(n) = 2T(n-1) + 3 $, with $ T(1) = 4 $. If the system evaluates $ T(m) $ for a specific $ m $, and the sum $ S(m) = \sum_{n=1}^{m} T(n) $ models total energy over time, compute $ S(5) $.

📌 Solution: We are given a recurrence:

📌 T(1) = 4, \quad T(n) = 2T(n-1) + 3 \text{ for } n \geq 2.

📌 We compute $ T(n) $ for $ n = 1 $ to $ 5 $:

📌 $ T(1) = 4 $

📌 $ T(2) = 2 \cdot 4 + 3 = 8 + 3 = 11 $

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