Question

The sum of three numbers in A.P. is $27$ and the sum of their squares is $275$. Find the numbers.

Answer 1

Let the three numbers are in AP$=a,a+d,a+2d$
According to the question,
The sum of three terms$=27$
$\Rightarrow a+(a+d)+(a+2d)=27$
$\Rightarrow 3a+3d=27$
$\Rightarrow a+d=9$
$\Rightarrow a=9-d....\left(i\right)$
and the sum of their squares$=275$
$\Rightarrow a^2+(a+d{)}^2+(a+2d{)}^2=275$
$\Rightarrow (9-d{)}^2+(9{)}^2+(9-d+2d{)}^2=275$ [from$\left(i\right)$]
$\Rightarrow 81+d^2-18d+81+81+d^2+18d=275$
$\Rightarrow 243+2d^2=275$
$\Rightarrow 2d^2=275-243$
$\Rightarrow 2d^2=32$
$\Rightarrow d^2=16$
$\Rightarrow d=\sqrt{16}$
$\Rightarrow d=\underset{–}{+}4$
Now, if $d=4,$ then $a=9-4=5$
and if $d=-4,$ then $a=9-(-4)=9+4=13$
So, the numbers are $\to$
if $a=5$ and $d=4$
$5,9,13$
and if$a=13$ and $d=-4$
$13,9,5$