Explanation

(c) When $f_{1}\ $and $f_{2}$ are focal lengths of lenses combined together, image formation takes place as follows
From lens formula
$\frac{1}{v'} - \frac{1}{u} = \frac{1}{f_{1}}\ \ \ \ \ \ \ \ldots\ldots\ldots\ldots(i)$
$\frac{1}{v} - \frac{1}{v'} = \frac{1}{f_{2}}\ \ \ \ \ \ \ \ \ \ \ \ldots\ldots..(ii)$
Adding Eqs. (i) and (ii), we get
$\frac{1}{v} - \frac{1}{u} = \ \frac{1}{f_{1}} + \frac{1}{f_{2}}$
If this lens is replaced by a single lens, then focal length of combination is
$\frac{1}{F} = \ \frac{1}{f_{1}} + \frac{1}{f_{2}} = \frac{1}{v} - \frac{1}{u}$
$\Longrightarrow F = \frac{f₁f₂}{f₁ + f₂}$