The sum of the first three terms of a G.P.
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 :

1. (1) $(-\infty,-9] \cup[3, \infty)$

2. (2) $[-3, \infty)$

3. (3) $(-\infty,-3] \cup[9, \infty)$

4. (4) $(-\infty, 9]$

Correct Option: , 3

Solution:

Let terms of G.P. be $\frac{a}{r}, a, a r$

$\therefore a\left(\frac{1}{r}+1+r\right)=S$$\ldots$ (i)

and $a^{3}=27$

$\Rightarrow a=3$…..(ii)

Put $a=3$ in eqn. (1), we get

$S=3+3\left(r+\frac{1}{r}\right)$

If $f(x)=x+\frac{1}{x}$, then $f(x) \in(-\infty,-2] \cup[2, \infty)$

$\Rightarrow 3 f(x) \in(-\infty,-6] \cup[6, \infty)$

$\Rightarrow 3+3 f(x) \in(-\infty,-3] \cup[9, \infty)$

Then, it concludes that

$S \in(-\infty,-3] \cup[9, \infty)$