Question

Three numbers $a,b,c$ are between $2$ and $18$ such that their sum is $25.2,a,b$ are in $A.P.$ and $b,c,$ $18$ are in $G.P.$ Find $abc$.

Answer 1

$a+b+c=25$ .....(1)

$2,a,binAP\Rightarrow 2+b=2a$ .....(2)
$b,c,18inGP\Rightarrow 18b=c^2$ .....(3)
Eliminating a from $\left(1\right),\left(2\right),3b+2c=48$ .....(4)

Eliminating b from $\left(3\right),\left(4\right),c2+12c=288$
$\Rightarrow c=12,–24$
But c lies between $2$ and $18.$

$\therefore c=12,b=8,a=5$

$\Rightarrow abc=\left(5\right)\left(8\right)\left(12\right)=480.$