Question

The sum of three numbers in A.P is $21$ and their product is $231.$ Find the numbers.

• A. $3,7,11$ or $11,7,3$
• B. $3,6,12$ or $12,6,3$
• C. $-3,-7,-11$ or $-11,-7,-3$
• D. $-3,-6,-12$ or $-12,-6,-3$
  

Answer 1

The correct option is A $3,7,11$ or $11,7,3$

Let the numbers be $(a-d),a,(a+d)$. Then,
Sum $=21\Rightarrow (a-d)+a+(a+d)=21\Rightarrow 3a=21\Rightarrow a=7$
Product $=231\Rightarrow (a-d)\times a\times (a+d)=231$
$\Rightarrow a(a^2-d^2)=231$
$\Rightarrow 7(49-d^2)=231$
$\Rightarrow (49-d^2)=33$
$\Rightarrow d^2=16\Rightarrow d=\pm 4$
If $d=4$, then the numbers are $3,7,11$. If $d=-4$, then the numbers are $11,7,3$
Thus, the numbers are $3,7,11$ or $11,7,3.$