Question

$x$ is one of the three consecutive numbers in G.P. whose sum is 38 and their product is 1728. The value of $x$ can be ___.

• A. 18
• B. 12
• C. 4
• D. 16

Answer 1

Answer: (A), (B)

The correct options are
A 18
B 12

Let the three numbers in G.P. are $\frac{a}{r},a,ar.$
Then, $\frac{a}{r}\times a\times ar=1728.$
$\Rightarrow a^3=1728$
$\Rightarrow a=12$
$\Rightarrow \frac{12}{r}+12+12r=38$
$\Rightarrow 12(\frac{1}{r}+r)=26$
$\Rightarrow 12(1+r^2)=26r$
$\Rightarrow 12r^2-26r+12=0$
$\Rightarrow 6r^2-13r+6=0$
$\Rightarrow 6r^2-9r-4r+6=0$
$\Rightarrow (3r-2)(2r-3)=0$
$\Rightarrow r=\frac{2}{3},\frac{3}{2}$
If $r=\frac{2}{3},$ then the numbers are 18, 12 and 8.
If $r=\frac{3}{2},$ then the numbers are 8, 12 and 18.