Question

The sum of three numbers in A.P. is $27$ and the sum of their squares is $293,$ find them.

Answer 1

Let the numbers be $a-d,a,a+d$
so $3a=27$
$\Rightarrow$ $a=9$
Also ${(a-d)}^2+a^2+{(a+d)}^2=293.$
$3a^2+2d^2=293$
$d^2=25$
$\Rightarrow$ $d=\pm 5$
therefore numbers are $4,9,14.$