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the guardians of the galaxy nebula - Network4

📅 May 19, 2026 👤 scraface

May 19, 2026

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📌 \( 120 = \frac{n}{2}(2×4 + (n−1)×4) = \frac{n}{2}(8 + 4n − 4) = \frac{n}{2}(4n + 4) = 2n(n + 1) \)

📌 So \( 2n^2 + 2n = 120 → n^2 + n − 60 = 0 \)

📌 \( n = \frac{-1 \pm \sqrt{1 + 240}}{2} = \frac{-1 \pm \sqrt{241}}{2} \) — not integer.

📌 Try: \( S_n = 90 \), a=5, d=5

📌 \( 90 = \frac{n}{2}(10 + (n−1)5) = \frac{n}{2}(5n + 5) = \frac{5n(n+1)}{2} \)

📌 \( 180 = 5n(n+1) → n(n+1) = 36 → n² + n − 36 = 0 → n = [-1±√145]/2 — not integer.

📌 Try: Sₙ = 60, a=3, d=3

📌 \( 60 = \frac{n}{2}(6 + (n−1)3) = \frac{n}{2}(3n + 3) = \frac{3n(n+1)}{2} \)

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