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Home » Use the formula: \(a_n = ar^n-1\). - ClickBalance

Use the formula: \(a_n = ar^n-1\). - ClickBalance

📅 May 18, 2026 👤 scraface

May 18, 2026

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📌 But $ \theta = 0 $ appears in both cases, so we list all distinct solutions:

📌 + \frac{2\pi}{5} + \frac{4\pi}{5} + \frac{6\pi}{5} + \frac{8\pi}{5} = \frac{(0 + 2 + 4 + 6 + 8)\pi}{5} = \frac{20\pi}{5} = 4\pi

📌 Thus, the sum of all such angles is $ \boxed{4\pi} $.

📌 Question: A historian of science analyzes an ancient diagram of celestial orbits, where the minimum value of the expression $ |2\sin x + 3\cos x + 4| $ over real $ x $ reflects a key symmetry in Ptolemaic model predictions. Find this minimum value.

📌 Solution: We aim to minimize:

📌 |2\sin x + 3\cos x + 4|

📌 First, consider the expression inside the absolute value: $ 2\sin x + 3\cos x $. This is a linear combination of sine and cosine. We write it in amplitude-phase form:

📌 2\sin x + 3\cos x = R\sin(x + \phi)

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