What am I doing wrong?, springs

Here's the question:

An arbitrary force is applied to two springs in series, and the question is, how is the overall spring constant for the two series springs related to the spring constants of the individual springs?

To review, the idea of spring constant is that the spring length deflects a distance (x) that is proportional to the applied force, F. The spring constant k is the constant of proportionality:

(1) F = kx

It is critical to realize that for two springs in series and a force applied, that at equilibrium the two springs must push on each other with the same force as was applied to the system; else they would not be in equilibrium:

(2) F = kx = k1x1 = k2x2, where

(3) x = x1 + x2

Now it simply remains to solve these two equations in two unknowns for the expression for k in terms of k1 and k2:

(4) k = f(k1, k2).

We can solve eqn 2 for either x1 or x2 in terms of the other; arbitrarily we solve for x1:

(5) x1 = k2x2/k1

We can then plug eqn 5 into eqn 3 to get:

(6) x = k2x2/k1 + x2,

which may be simplified into

(7) x = x2 (k2/k1 + 1) = x2 (k2 + k1)/k1

And we know from eqn 2 that

(8) kx = k2x2,

so we can plug eqn 7 into eqn 8 to get:

(9) k x2 (k2 + k1)/ k1 = k2 x2

In eqn 9, the variable x2 cancels on both sides, leaving

(10) k = k2 k1 / (k2 + k1)

From which the more familiar form follows:

(11) 1/k = 1/k1 + 1/k2

Again, the critical realization necessary to solve this problem is that when the springs are in series and a force is applied, that the same force is applied to each spring, and the two springs apply that same force on each other to maintain the new equilibrium state.

In my opinion the solution is:

F/k1=x1

F/k2=x2

the total displacement is

X=x1+x2

The resulting K is

K=F/(x1+x2)= F/(F/k1+F/k2)= k1*K2/(k1+k2)