Two massless springs of spring constant \(K_1\) and \(K_2\) are connected one after the other forming a single chain, suspended vertically and certain mass is attached to the free end. If \(e_1\) and \(e_2\) are their respective extensions and \(f\) is their stretching force, the total extension produced is

The total restoring force is related to extension as follows:
\(f=-K x \quad---(1)\)
The extension in the springs are related to the restoring force as follows:
\(f=-K_1 e_1---(2)\)
and
\(f=-K_2 e_2---(3)\)
The total extension of the springs is:
\(x=e_1+e_2---(4)\)
Using equation (1),(2) and (3)
\(\begin{aligned} & \Rightarrow\left(-\frac{f}{K}\right)=\left(-\frac{f}{K_1}\right)+\left(-\frac{f}{K_2}\right) \\ & \Rightarrow \frac{1}{K}=\left(\frac{1}{K_1}+\frac{1}{K_2}\right)---(5)\end{aligned}\)
The total extension using equation (4) and (5)is given by,
\(x=\frac{f}{K}=f\left(\frac{1}{K_1}+\frac{1}{K_2}\right)\)