Rate of transmission of heat
$=\frac{Temperature\,difference}{Thermal\,Resistance}$
$\therefore \frac{dQ}{dt}=\frac{d\theta}{R}$
Here, $\frac{dQ}{dt}=\frac{\left(\theta-\theta_{2}\right)}{R_{2}}=\frac{\theta_{1}-\theta }{R_{1}}$
$\Rightarrow \frac{\theta -\theta_{2}}{R_{2}}=\frac{\theta _{1}-\theta }{R_{1}}$
$\Rightarrow R_{1}\theta-R_{1}\theta_{2}=R_{2}\theta_{1}-R_{2}\theta$
$\Rightarrow \theta\left(R_{1}+R_{2}\right)=R_{2}\theta_{1}+R_{1}\theta_{2}$
$\therefore \theta=\frac{\left(R_{2}\theta_{1}+R_{1}\theta_{2}\right)}{\left(R_{1}+R_{2}\right)}$