JEE PYQ: Sequence And Series Question 18
Question 18 - 2020 (02 Sep Shift 1)
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) $(-\infty, -9] \cup [3, \infty)$
(2) $[-3, \infty)$
(3) $(-\infty, -3] \cup [9, \infty)$
(4) $(-\infty, 9]$
Show Answer
Answer: (3)
Solution
Let terms of G.P. be $\frac{a}{r}$, $a$, $ar$
$\therefore a\left(\frac{1}{r} + 1 + r\right) = S$ …(i)
and $a^3 = 27$
$\Rightarrow a = 3$
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 $f(x) \in (-\infty, -2] \cup [2, \infty)$
$\Rightarrow 3f(x) \in (-\infty, -6] \cup [6, \infty)$
$\Rightarrow 3 + 3f(x) \in (-\infty, -3] \cup [9, \infty)$
Then, it concludes that
$S \in (-\infty, -3] \cup [9, \infty)$
BETA
