Springs of force constants K, 2K, 4K, 8K, 16K,… are connected in series. The mass ‘m’ kg is attached to the lower end of the last spring and the system is allowed to vibrate. The frequency of oscillation is

K, 2K, 4K, 8K, 16K,… are connected in series

effective spring constant is $K_{effective}=K_{eff}$ is
$\frac{1}{{\text{K}}_{\text{eff}}}=\frac{\text{1}}{\text{K}}\text{+}\frac{\text{1}}{\text{2K}}\text{+}\frac{\text{1}}{\text{4K}}+---$
$\frac{1}{{\text{K}}_{\text{eff}}}=\frac{\text{1}}{\text{K}}\left[1+\frac{1}{2}+\frac{1}{2^2}+\mathrm{...}\right]$ , the paranthesis part is in geometric progression 

whose first term =1 and common ratio is r=$\frac{1}{2}$

$$\begin{gathered} sum to infinite terms is S_{\infty }=\frac{a}{1-r} \\ where a is first term, r is common ratio \end{gathered}$$
  $\frac{\text{1}}{{\text{K}}_{eff}}=\frac{\text{1}}{\text{K}}\left[\frac{\text{a}}{\text{1-r}}\right]=\frac{\text{1}}{\text{K}}\left[\frac{1}{\text{1-}{\displaystyle \frac{1}{2}}}\right]$
$\frac{1}{{\text{K}}_{\text{eff}}}=\frac{\text{1}}{\text{K}}\left[\frac{2}{\text{1}}\right]$

$K_{eff}=\frac{K}{2}$
Frequency   $\eta \text{=}\frac{\text{1}}{\text{2π}}\sqrt{\frac{{{\text{K}}_{\text{ef}}}_{\text{f}}}{\text{m}}}=\frac{\text{1}}{\text{2π}}\sqrt{\frac{\frac{\text{K}}{\text{2}}}{\text{m}}}$ 

$due to series of springs new frequency is=\frac{\text{1}}{\text{2π}}\sqrt{\frac{\text{K}}{\text{2m}}}$