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hyatt place new york yonkers - AssociationVoting

📅 May 17, 2026 👤 scraface

May 17, 2026

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📌 a_{34} = 2 + 33 \cdot 3 = 101 > 100

📌 Thus, there are \(\boxed{33}\) terms before the value exceeds 100.

📌 Question:** The mechanical engineer is calibrating sensors placed at positions \( y \) such that \( y \) is a positive multiple of 5 and \( y^3 < 12000 \). What is the largest possible value of \( y \)?

📌 Solution:** We are given that \( y \) is a positive multiple of 5 and \( y^3 < 12000 \). Solve the inequality:

📌 y^3 < 12000

📌 Take cube roots of both sides:

📌 y < \sqrt[3]{12000}

📌 22^3 = 10648,\quad 23^3 = 12167

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Swap \( x \) and \( y \): \( x = rac{2y + 3}{y - 1} \).
Cross-multiply: \( x(y - 1) = 2y + 3 \).
Expand: \( xy - x = 2y + 3 \).
Rearrange: \( xy - 2y = x + 3 \).
Factor: \( y(x - 2) = x + 3 \).
Solve for \( y \): \( y = rac{x + 3}{x - 2} \).
#### Inverse: \( f^{-1}(x) = rac{x + 3}{x - 2} \)
Evaluate the limit \( \lim_{x o 2} rac{x^2 - 4}{x - 2} \).
Factor the numerator: \( x^2 - 4 = (x - 2)(x + 2) \).
The expression becomes \( rac{(x - 2)(x + 2)}{x - 2} \).

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