Question

The sum of three numbers of A.P. is $27$ and their product is $504$, then the numbers are :

• A. $4,8,14$
• B. $3,8,14$
• C. $4,7,15$
• D. $4,9,14$

Answer 1

The correct option is D $4,9,14$

Let, three numbers $a-d,a,a+d$ are in AP.
Given, $a-d+a+a+d=27$

$\Rightarrow 3a=27$

$\Rightarrow a=9$
Also given, $(a-d)a(a+d)=504$
$\Rightarrow a(a^2-d^2)=504$
Now, put $a=9$
$\Rightarrow 9(9^2-d^2)=504$
$\Rightarrow 9^2-d^2=56$
$\Rightarrow 81-d^2=56$
$\Rightarrow d^2=81-56$
$\Rightarrow d^2=25$

$\Rightarrow d=5$ or $d=-5$
Put $a=9,d=5$ in given numbers. we get,
$9-5,9,9+5=4,9,14$
Again put $a=9,d=-5$ in given numbers. we get
$9+5,9,9-5=14,9,4$
Option D is correct.