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Home » The longest side is \( 5x = 25 \) cm. - Treasure Valley Movers

The longest side is \( 5x = 25 \) cm. - Treasure Valley Movers

📅 April 16, 2026 👤 scraface

Apr 16, 2026

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📌 Question: A science administrator allocates research grants in cycles of 12 months. If a particular grant begins in month $ m $, and the $ n $-th grant issuance follows the pattern $ m + 12(n - 1) $, what is the smallest positive integer $ m $ such that the 10th grant is issued in month number 84?

📌 Solution: The month of the $ n $-th grant is given by:

📌 We are told that the 10th grant is issued in month 84. Substituting $ n = 10 $:

📌 But $ m $ must be a positive integer. Since the pattern repeats every 12 months, we find the equivalent positive month by computing modulo 12. However, the grant issuance month is tied to $ m $, which must be in the range $ 1 $ to $ 12 $. The general solution is:

📌 Solution: This is an arithmetic sequence with first term $ a = 2 $, common difference $ d = 4 $. The sum of the first $ n $ terms is given by:

📌 The largest integer $ n $ satisfying this is $ n = 9 $.

📌 Solution: This is an arithmetic sequence with $ a = 5 $, $ d = 8 $, and $ x < 100 $. The $ n $-th term is:

📌 Question: A science administrator reviews funding cycles where each grant is issued every 7 years starting in year 3. What is the sum of all years (positive integers) in which a grant is issued, between year 1 and year 100?

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