Sequences And Series Ques 6
- If $a_1, a_2, \ldots, a_n$ are positive real numbers whose product is a fixed number $c$, then the minimum value of $a_1+a_2+\ldots+a_{n-1}+2 a_n$ is
$(2002,1 \mathrm{M})$
(a) $n(2 c)^{1 / n}$
(b) $(n+1) c^{1 / n}$
(c) $2 n c^{1 / n}$
(d) $(n+1)(2 c)^{1 / n}$
Passage Based Problems
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Answer:
Correct Answer: 6.(a)
Solution: (a) Given, $a_1 a_2 a_3 \ldots a_n=c$
$ \begin{aligned} & \Rightarrow \quad a_1 a_2 a_3 \ldots\left(a_{n-1}\right)\left(2 a_n\right)=2 c \quad ……..(i)\\ & \therefore \quad \frac{a_1+a_2+a_3+\ldots+2 a_n}{n} \geq\left(a_1 \cdot a_2 \cdot a_3 \ldots 2 a_n\right)^{1 / n} \end{aligned} $
[using AM $\geq \mathrm{GM}$ ]
$\Rightarrow \quad a_1+a_2+a_3+\ldots+2 a_n \geq n(2 c)^{1 / n}\quad $ [from Eq. (i)]
$\Rightarrow \quad $ Minimum value of
$ a_1+a_2+a_3+\ldots+2 a_n=n(2 c)^{1 / n} $
BETA
