Two wires of resistances $R_1$ and $R_2$ have temperature coefficient of resistances ${\alpha }_1$ and ${\alpha }_2$ respectively . These are joined in series. The effective temperature coefficient of resistance is 

At temperature  $\theta$

$R_1^1=R_1\left[1+{\alpha }_1\Delta \theta \right], R_2^1=R_2\left[1+{\alpha }_2\Delta \theta \right]$

At  $0°C$ ,  ${\mathrm{R}}_{\text{eq}}={\text{R}}_{\text{1}}{\text{+R}}_{\text{2}}$

At  $\theta °C$ ,  $R_{eq}^1=R_1^1+R_2^1=R_1\left[1+{\alpha }_1\theta \right]+R_2\left[1+{\alpha }_2\theta \right]$

Let equivalent temperature coefficient is  ${\alpha }^1$

At  $\theta °C$ ,  $R_{eq}^1=R_{eq}\left[1+{\alpha }^1\theta \right]$

$\left(R_1+R_2\right)\left[1+{\alpha }^1\theta \right]=R_1\left[1+{\alpha }_1\theta \right]+R_2\left[1+{\alpha }_2\theta \right]$

${\alpha }^1=\frac{R_1{\alpha }_1+R_2{\alpha }_2}{R_1+R_2}$