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Home » The sum of an infinite geometric series is 12, and the first term is 3. Find the common ratio. - Upplift

The sum of an infinite geometric series is 12, and the first term is 3. Find the common ratio. - Upplift

📅 March 2, 2026 👤 scraface

Mar 02, 2026

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📌 Find the derivative of \( y = \ln(x^2 + 1) \) at \( x = 1 \).

📌 Use chain rule: \( rac{dy}{dx} = rac{1}{x^2 + 1} \cdot 2x = rac{2x}{x^2 + 1} \).

📌 Evaluate at \( x = 1 \): \( rac{2(1)}{1^2 + 1} = rac{2}{2} = 1 \).

📌 The sum of an infinite geometric series is 12, and the first term is 3. Find the common ratio.

📌 Sum formula: \( S = rac{a}{1 - r} \), so \( 12 = rac{3}{1 - r} \).

📌 Solve: \( 12(1 - r) = 3 \) â \( 12 - 12r = 3 \) â \( 12r = 9 \) â \( r = rac{3}{4} \).

📌 A right triangle has legs of lengths \( \sqrt{5} \) and \( \sqrt{11} \). Find the length of the altitude to the hypotenuse.

📌 Hypotenuse: \( c = \sqrt{(\sqrt{5})^2 + (\sqrt{11})^2} = \sqrt{5 + 11} = \sqrt{16} = 4 \).

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