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how do you mate villagers - DevRocket

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 A sequence is defined recursively by \( a_1 = 2 \) and \( a_{n} = 3a_{n-1} + 4 \) for \( n \geq 2 \). What is \( a_4 \)?

📌 \( a_2 = 3 \times 2 + 4 = 10 \)

📌 \( a_3 = 3 \times 10 + 4 = 34 \)

📌 \( a_4 = 3 \times 34 + 4 = 106 \)

📌 The sum of the first \( n \) terms of an arithmetic series is given by \( S_n = \frac{n}{2}(2a + (n-1)d) \). If the first term \( a = 5 \) and the common difference \( d = 3 \), what is the sum of the first 10 terms?

📌 \( S_{10} = \frac{10}{2}(2 \times 5 + (10-1) \times 3) \)

📌 \( S_{10} = 5(10 + 27) = 5 \times 37 = 185 \)

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