Question:
The sum of the first three terms of a G.P. is $S$ and their product is 27 . Then all such S lie in :
Correct Option: , 4
Solution:
Let three terms of G.P. are $\frac{\mathrm{a}}{\mathrm{r}}, \mathrm{a}$, ar
product $=27$
$\Rightarrow a^{3}=27 \Rightarrow a=3$
$\mathrm{S}=\frac{3}{\mathrm{r}}+3 \mathrm{r}+3$
For $r>0$
$\frac{\frac{3}{r}+3 r}{2} \geq \sqrt{3^{2}} \quad($ By $\mathrm{AM} \geq \mathrm{GM})$
$\Rightarrow \frac{3}{r}+3 r \geq 6$........(1)
For $r<0 \quad \frac{3}{r}+3 r \leq-6$ $\ldots(2)$
From (1) & (2)
$S \in(-\infty-3] \cup[9, \infty]$
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