<<...an equation that shows why the spring is
stiffer if pre-loaded. >>

Not only that, but you have a basic law of physics on your side:
Conservation of Energy.  If it takes energy to compress the spring and
insert a spacer, then that energy needs to go somewhere--is stored
within the coils of the spring.  The inertial energy used to compress
the spring becomes stored (techncially, "potential") energy within the
coils.  The energy that you spent compressing the spring (watch those
fingers!) is now pushing back just as hard.  This translates to a higher
spring rate.

Barry Armitage