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"Blood Moon Ursaluna: The Cosmic Foreboding Everyone’s Talking About! - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 The sum of the first \( n \) terms of an arithmetic sequence is given by \( S_n = \frac{n}{2}(2a + (n-1)d) \). If the first term is 3 and the common difference is 2, find \( n \) such that \( S_n = 210 \).

📌 Substitute: \( 210 = \frac{n}{2}(2 \times 3 + (n-1) \times 2) = \frac{n}{2}(6 + 2n - 2) = \frac{n}{2}(2n + 4) \).

📌 Simplify: \( 210 = n(n + 2) \) → \( n^2 + 2n - 210 = 0 \).

📌 Solve quadratic: \( n = \frac{-2 \pm \sqrt{4 + 840}}{2} = \frac{-2 \pm 29.02}{2} \).

📌 Positive root: \( n = \frac{27.02}{2} \approx 13.51 \), so \( n = 14 \) (next whole number).

📌 Verify: \( S_{14} = \frac{14}{2}(6 + 26) = 7 \times 32 = 224 \) → too high.

📌 Try \( n = 13 \): \( S_{13} = \frac{13}{2}(6 + 24) = 6.5 \times 30 = 195 \) → too low.

📌 Recheck equation: \( n(n + 2) = 210 \) → \( n^2 + 2n - 210 = 0 \).

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