Two thin lenses of focal length $\displaystyle \mathrm{f}_{1}$ and $\displaystyle \mathrm{f}_{2}$ are placed in contact with each other coaxially. Prove that the focal length f of the combination is given by $\displaystyle \mathrm{f}=\frac{\mathrm{f}_{1} \mathrm{f}_{2}}{\mathrm{f}_{1}+\mathrm{f}_{2}}$.

Proving the focal length of combination is $\displaystyle \mathrm{f}=\frac{\mathrm{f}_{1} \mathrm{f}_{2}}{\mathrm{f}_{1}+\mathrm{f}_{2}}$ $\displaystyle 2$ \[\xrightarrow[\mathrm{O}]{ } \] For the image formed by lens A \[\frac{1}{\mathrm{v}_{1}}-\frac{1}{u}=\frac{1}{\mathrm{f}_{1}} \] For the image formed by second lens B \[\frac{1}{\mathrm{v}}-\frac{1}{\mathrm{v}_{1}}=\frac{1}{\mathrm{f}_{2}} \] Adding (i)and(ii) \[\begin{aligned} & \frac{1}{\mathrm{v}}-\frac{1}{u}=\frac{1}{\mathrm{f}_{2}}+\frac{1}{\mathrm{f}_{1}} \\ & \frac{1}{\mathrm{f}}=\frac{1}{\mathrm{f}_{1}}+\frac{1}{\mathrm{f}_{2}} \\ & \mathrm{f}=\frac{\mathrm{f}_{1} \mathrm{f}_{2}}{\mathrm{f}_{1}+\mathrm{f}_{2}} \end{aligned} \]