Mathematics - Sequence & Series Question with Solution | TestHub

Write $a,b,c,d=u,u+t,u+2t,u+3t$ with nonzero integer $t$. Then $3u^2+(6t-1)u+5t^2-3t=0$. Its discriminant is $-24t^2+24t+1\ge 0$, forcing $t=1$ (since $t\ne 0$ ). The resulting quadratic is $3u^2+5u+2=0$, with only integer root $u=-1$. The terms are (-1, 0, 1, 2) and sum to 2.