Question

Find the sum of the three numbers in G.P whose product is 216 and the sum of the products of them taken in pairs is 126:

• A. 28
• B. 21
• C. $\frac{35}{4}$
• D. None of these

Answer 1

The correct option is B 21

$a.b.c=a.ar.a{r}^2=216$
$\Rightarrow b=ar=6$ ...(1)
and $ab+bc+ac=a.ar+ar.a{r}^2+a.a{r}^2=126$
$\Rightarrow a^{2}r+a^2{r}^3+a^2{r}^2=126$
$\Rightarrow a^{2}r+a^2{r}^3=90$ $(\because ar=6)$
$\Rightarrow 6a+36r=90$
$\Rightarrow a+6r=15$ ...(2)
from equation (1) and (2), we get
$\frac{6}{r}+6r=15$
$\Rightarrow 6r^2-15r+6=0$
$\Rightarrow r=2or\frac{1}{2}$
$\therefore a,ar,a{r}^2=3,6,12or12,6,3$
$\therefore a+b+c=21$