Derive the thin lens equation.

(N/A) For refraction at two spherical surfaces of a lens,the general equation is:
$\frac{n_{1}}{-u} + \frac{n_{1}}{v} = (n_{2} - n_{1}) \left[ \frac{1}{R_{1}} - \frac{1}{R_{2}} \right]$
Dividing both sides by $n_{1}$,we get:
$\frac{1}{-u} + \frac{1}{v} = \left( \frac{n_{2}}{n_{1}} - 1 \right) \left[ \frac{1}{R_{1}} - \frac{1}{R_{2}} \right]$
Since $n_{21} = \frac{n_{2}}{n_{1}}$,we have:
$\frac{1}{v} - \frac{1}{u} = (n_{21} - 1) \left[ \frac{1}{R_{1}} - \frac{1}{R_{2}} \right] \quad \dots (1)$
The lens maker's formula is given by:
$\frac{1}{f} = (n_{21} - 1) \left[ \frac{1}{R_{1}} - \frac{1}{R_{2}} \right] \quad \dots (2)$
By comparing equations $(1)$ and $(2)$,we obtain the thin lens equation:
$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$
Here,the distances are measured according to the sign convention,and this equation is valid for both real and virtual images formed by convex and concave lenses.