$ E_B - E_A = \frac{hc}{\lambda_2}$

$ E_C - E_B = \frac{hc}{\lambda_1}$

$ E_C - E_A = \frac{hc}{\lambda_3}$

$ E_C - E_A = (E_C - E_B)+(E_B - E_A)$

$\Rightarrow \frac{hc}{\lambda_3} = \frac{hc}{\lambda_1} + \frac{hc}{\lambda_2}$

$\Rightarrow \frac{1}{\lambda_3} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2}$

$\Rightarrow \lambda_3 = \frac{\lambda_1\lambda_2} {\lambda_1+\lambda_2}$