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8. Sequences and Series
medium
The sum of the first three terms of a $G.P.$ is $S$ and their product is $27 .$ Then all such $S$ lie in
A
$[-3, \infty)$
B
$(-\infty, 9]$
C
$(-\infty,-9] \cup[3, \infty)$
D
$(-\infty,-3] \cup[9, \infty)$
(JEE MAIN-2020)
Solution
Let three terms of G.P. are $\frac{\mathrm{a}}{\mathrm{r}}, \mathrm{a}, \mathrm{ar}$ product $=27$
$\Rightarrow \mathrm{a}^{3}=27 \Rightarrow \mathrm{a}=3$
$S=\frac{3}{r}+3 r+3$
For ${r}>0$
$\frac{\frac{3}{r}+3 r}{2} \geq \sqrt{3^{2}} \quad($ By $A M \geq G M)$
$\Rightarrow \frac{3}{r}+3 r \geq 6$
For $r<0 \quad \frac{3}{r}+3 r \leq-6 \quad \ldots(2)$
From ( 1) (2)
$\mathrm{S} \in(-\infty-3) \cup[9, \infty]$
Standard 11
Mathematics
