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Home » Solution: Convert $ 314_7 $ to base 10: - go-checkin.com

Solution: Convert $ 314_7 $ to base 10: - go-checkin.com

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Question: A bioinformatician studies a DNA sequence where the probability of a mutation at position $ n $ is $ p_n = \frac{n}{n^3 + 2n} $. Simplify $ p_n $ and find its limit as $ n \to \infty $.

📌 Solution: Simplify $ p_n = \frac{n}{n(n^2 + 2)} = \frac{1}{n^2 + 2} $. As $ n \to \infty $, $ n^2 + 2 \to \infty $, so $ \lim_{n \to \infty} p_n = \boxed{0} $.

📌 Question: A civil engineer calculates the stress on a beam with cross-section $ A = \frac{4}{x} - x^2 $, where $ x > 0 $. Find the maximum stress.

📌 Solution: Take derivative $ A'(x) = -\frac{4}{x^2} - 2x $. Set $ A'(x) = 0 $: $ -\frac{4}{x^2} - 2x = 0 \Rightarrow -4 - 2x^3 = 0 \Rightarrow x^3 = -2 $. No positive roots. Analyze behavior: as $ x \to 0^+ $, $ A \to \infty $; as $ x \to \infty $, $ A \to -\infty $. Thus, no maximum exists; the function increases without bound near $ x = 0 $. However, if constrained, recheck calculations. Correct approach:

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