The coefficient of linear expansion of a rod changes from $\;\alpha_{1}\;$ to $\;\alpha_{2}\;$ from end to end for a length l of the rod . If the temperature changes from $\;0^{0}C\;$ to $\;t^{0}C\;$ , the change of length is

Question

$(a)\;\large\frac{(\alpha_{2}-\alpha_{1})}{2}\;lt\qquad(b)\;\large\frac{(\alpha_{1}+\alpha_{2})}{2}\;lt\qquad(c)\;\sqrt{\alpha_{1} \alpha_{2}}\qquad(d)\;\large\frac{(\alpha_{1}-\alpha_{2})}{2}\;lt$