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Home » We seek the smallest three-digit integer \( D \) such that: - Groen Casting

We seek the smallest three-digit integer \( D \) such that: - Groen Casting

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: Let the arithmetic sequence have first term $ a_1 = 15 $ and fifth term $ a_5 = 35 $. The common difference $ d $ satisfies $ a_5 = a_1 + 4d $. Substituting: $ 35 = 15 + 4d \Rightarrow d = 5 $. The sum of the first $ n = 10 $ terms is $ S_{10} = rac{10}{2} [2(15) + 9(5)] = 5(30 + 45) = 5(75) = 375 $. oxed{375}

📌 Question: A science educator designs a game where each correct challenge awards $ x $ points and a 5-point bonus for every 3 challenges completed. If a student earns 68 points after 10 challenges, what is $ x $?

📌 Solution: Let $ x $ be points per correct challenge. For 10 challenges, there are $ \lfloor rac{10 - 1}{3}

📌 floor = 3 $ bonuses (after challenges 3, 6, 9). Total points: $ 10x + 3(5) = 68 \Rightarrow 10x + 15 = 68 \Rightarrow 10x = 53 \Rightarrow x = 5.3 $. oxed{5.3}

📌 Question: A middle school studentâs bridge model uses 4 support beams for every 3 meters of span. If the bridge is 18 meters long, how many beams are required, assuming beams are spaced at 1.5-meter intervals?

📌 Solution: The number of intervals is $ rac{18}{1.5} = 12 $. Since beams are placed at both ends, total beams = $ 12 + 1 = 13 $. However, the problem states 4 beams per 3 meters, which implies $ rac{4}{3} $ beams per meter. For 18 meters: $ rac{4}{3} imes 18 = 24 $. oxed{24}

📌 Question: In a science game, a playerâs score doubles every level. If the score at level 3 is 128, what is the score at level 5?

📌 Solution: Let the score at level $ n $ be $ S_n = S_1 \cdot 2^{n-1} $. Given $ S_3 = S_1 \cdot 4 = 128 \Rightarrow S_1 = 32 $. Then $ S_5 = 32 \cdot 2^{4} = 32 \cdot 16 = 512 $. oxed{512}

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