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Mathematics - Sequence & Series Question with Solution | TestHub

MathematicsSequence & SeriesMeans / inequalityMedium2 minQB

MathematicsMediummultiple choice

Let $0 < a < b$ such that $2, 2^{x} - \frac{7}{2}, 2^{x} - 5$ are the harmonic mean, arithmetic mean and geometric mean of $a, b$ respectively. Then which of the following option(s) is/are correct?

Options:(select one or more)

Answer:

B, C

Solution:

$(GM)^{2} = AM \times HM : (2^{x} - 5)^{2} = 2 (2^{x} - \frac{7}{2}) \Rightarrow 2^{2 x} - 12 \cdot 2^{x} + 32 = 0 \Rightarrow 2^{x} = 8$ or 4. $2^{x} = 4$ gives $GM = - 1 < 0$ , rejected. So $x = 3$ only. Then $a + b = 2 \cdot 4.5 = 9$ and $ab = 3^{2} = 9$ , so $a^{2} + b^{2} = 81 - 18 = 63$ .

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:Means / inequality

⏱ 2mℹ️ Source: QB

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