Spring constant in mass spring combinations

[mrcotton]

Homework Statement

How do we derive the total spring constants of different mass spring systems

Homework Equations

F= kΔx

The Attempt at a Solution

 

[mrcotton]

I have just realised this is similar to this thread
https://www.physicsforums.com/showthread.php?t=711957
I was told to treat this as F1 = F2 and the displacements are equal.
I am struggling to convince myself that F(total) = F1=F2
any help gratefully received

 

[mrcotton]

In the parrallel case I can see how the displacement is the same but again I am confused by the forces.
I guess I need a couple of spring balances