If $a,b,c$  are non – zero real numbers such that $3\left(a^2+b^2+c^2+1\right)=2\left(a+b+c+ab+bc+ca\right)$  then $a,b,c$  are in

$\therefore$  $\left[{\left(a-b\right)}^2+{\left(b-c\right)}^2+{\left(c-a\right)}^2\right]+\left[{\left(a-1\right)}^2+{\left(b-1\right)}^2+{\left(c-1\right)}^2\right]=0$

$\Rightarrow a-b=b-c=c-a=0$  and $a-b=b-1=c-1=0$

$\Rightarrow a=b=c=1$

$\Rightarrow a,b,c$  are in $A.P$  and $G.P$ . Also they are in $H.P$