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how to get rid of bacne - Malaeb

📅 May 4, 2026 👤 scraface

May 04, 2026

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📌 Solution: We are given the function $ P(d) = 3d + 7 $, and we are to compute the total number of microplastic particles from depth $ d = 1 $ to $ d = 10 $. This requires summing $ P(d) $ over integer values from 1 to 10:

📌 Question: A geographer analyzing coastal erosion maps uses remote sensing data and finds that the area of lost land over 8 consecutive years forms an arithmetic sequence. If the total area lost over the 8 years is 448 square kilometers and the sum of the areas lost in the first and eighth years is 112 square kilometers, what is the common difference of the sequence?

📌 Solution: Let the first term of the arithmetic sequence be $ a $ and the common difference be $ d $. The sequence has 8 terms, so the sum $ S_8 $ is:

📌 The eighth term is $ a + 7d $, and the first term is $ a $, so their sum is:

📌 Now solve the system. From equation (2): $ 2a + 7d = 112 $. Multiply both sides by 4:

📌 This matches equation (1), so the equations are consistent. Now use $ 2a + 7d = 112 $ to express $ a $ in terms of $ d $:

📌 For $ a $ to be real and $ d $ to be rational (and likely integer in such models), $ d $ must be even. Let $ d = 2k $. Then:

📌 Now verify sum: $ 8a + 28d = 8(56 - 7k) + 28(2k) = 448 - 56k + 56k = 448 $, which holds.

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