Justification for spring in series/parallel

[RubinLicht]

Homework Statement

Find the frequency of oscillation of mass m suspended by two springs having constants k1 and k2, in each of the configurations shown.

my question isn't on how to solve the problem but rather why the constants add with reciprocal action for parallel. I was able to figure this out with just logic and confirmed it with the internet, but i would like a formal "proof", if you can call it that, for why it works.
assume massless springs.

Homework Equations

F=-kx
Series: 1/k1+1/k2=1/Keq

The Attempt at a Solution

nada, already solved the problem, just want justification. Thanks for the help

EDIT: i am reading the proof on wikipedia, but it'll still take a while for me to get it as my physical intuition is still trying to catch up with the math. Parallel is easy, got that. all that's left is series

 

[RubinLicht]

Took me a while, but i got it. thanks all.