Sequences And Series Ques 28
- 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
(2011)
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Answer:
Correct Answer: 28.$(8)$
Solution: Using $\mathrm{AM} \geq \mathrm{GM}$,
$ \begin{aligned} & \frac{a^{-5}+a^{-4}+a^{-3}+a^{-3}+a^{-3}+1+a^8+a^{10}}{8} \\ & \geq\left(a^{-5} \cdot a^{-4} \cdot a^{-3} \cdot a^{-3} \cdot a^{-3} \cdot 1 \cdot a^8 \cdot a^{10}\right)^{\frac{1}{8}} \\ & \Rightarrow a^{-5}+a^{-4}+3 a^{-3}+1+a^8+a^{10} \geq 8 \cdot 1 \end{aligned} $
Hence, minimum value is $8$ .
BETA
