Mathematics - Sequence & Series Question with Solution | TestHub

MathematicsSequence & SeriesMeans / inequalityMedium2 minPYQ_2011

MathematicsMediumnumerical

The minimum value of the sum of real numbers $a^{- 5}, a^{- 4}, 3 a^{- 3}, 1, a^{8}$ and $a^{10}$ with $a > 0$ is

Answer:

8.00

Solution:

Using $A M \geq GM$ , $\frac{a ^{- 5} + a ^{- 4} + a ^{- 3} + a ^{- 3} + a ^{- 3} + 1 + a ^{8} + a ^{10}}{8} \geq (a^{- 5} \cdot a^{- 4} \cdot a^{- 3} \cdot a^{- 3} \cdot a^{- 3} \cdot 1 \cdot a^{8} \cdot a^{10})^{\frac{1}{8}} \Rightarrow a^{- 5} + a^{- 4} + 3 a^{- 3} + 1 + a^{8} + a^{10} \geq 8 \cdot 1$ Hence, minimum value is 8 .

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

⏱ 2mℹ️ Source: PYQ_2011

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