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Factor out 9 from \( x \)-terms: - Portal da AcÃºstica

📅 February 28, 2026 👤 scraface

Feb 28, 2026

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📌 \sum_{k=1}^{24} 3^k = 3 \cdot \frac{3^{24} - 1}{3 - 1} = \frac{3}{2} (3^{24} - 1)

📌 But working modulo $7$, we can compute the sum directly using cyclicity. First, find the pattern of $3^k \mod 7$:

📌 3^1 &\equiv 3 \mod 7 \\

📌 3^2 &\equiv 9 \equiv 2 \mod 7 \\

📌 3^3 &\equiv 3 \cdot 2 = 6 \mod 7 \\

📌 3^4 &\equiv 3 \cdot 6 = 18 \equiv 4 \mod 7 \\

📌 3^5 &\equiv 3 \cdot 4 = 12 \equiv 5 \mod 7 \\

📌 3^6 &\equiv 3 \cdot 5 = 15 \equiv 1 \mod 7 \\

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