Answer
$K_T=\frac{K_1K_2}{K_1+K_2}$
Work Step by Step
We know that
$F=K_T s$
This can be rearranged as:
$s=\frac{F}{K_T}$
Similarly for the first spring $s_1=\frac{F}{K_1}$
and for the second spring $s_2=\frac{F}{K_2}$
As $s=s_1+s_2$
$\implies \frac{1}{K_T}=\frac{1}{K_1}+\frac{1}{K_2}$
$\implies \frac{1}{K_T}=\frac{K_1+K_2}{K_1K_2}$
This can be rearranged as:
$\implies K_T=\frac{K_1K_2}{K_1+K_2}$
