Two radioactive elements A and B initially have same number of atoms. The half life of A is same as the average life of B. If $\lambda_{A}$ and $\lambda_{B}$ are decay constants of A and B respectively, then choose the correct relation from the given options.

<p>We are given that the half-life of A is the same as the average life of B. The relationship between half-life ($T_{1/2}$) and the decay constant ($\lambda$) is:</p> <p>$T_{1/2} = \frac{\ln 2}{\lambda}$</p> <p>For the average life ($\tau$), the relationship with the decay constant is:</p> <p>$\tau = \frac{1}{\lambda}$</p> <p>According to the given information, the half-life of A is equal to the average life of B:</p> <p>$T_{1/2(A)} = \tau_{B}$</p> <p>Now, we can substitute the relationships for half-life and average life:</p> <p>$\frac{\ln 2}{\lambda_{A}} = \frac{1}{\lambda_{B}}$</p> <p>To find the correct relationship between $\lambda_{A}$ and $\lambda_{B}$, we can rearrange the equation:</p> <p>$\lambda_{A} = \lambda_{B} \ln 2$</p>