Springs of spring constants K, 2K, 4K, 8K…..2048K are connected in series. A mass m is attached to one end of the system is allowed to oscillate. It’s time period is approximately $2\mathrm{\pi }\sqrt{\frac{\mathrm{Pm}}{\mathrm{K}}}$  Find the value of P.
 

$\begin{array}{l}\frac{1}{{\mathrm{k}}_{\mathrm{eff}}}=\frac{1}{\mathrm{k}}+\frac{1}{2\mathrm{k}}+\frac{1}{4\mathrm{k}}+.......+\frac{1}{2048\mathrm{k}}\\ =\frac{1}{\mathrm{k}}[1+\frac{1}{2}+\frac{1}{4}+.........]\\ {\mathrm{s}}_{\mathrm{\infty }}=\frac{\mathrm{a}}{1-\mathrm{r}} \mathrm{and} \mathrm{a}=1,\mathrm{r}=\frac{1}{2}\\ \frac{1}{{\mathrm{k}}_{\mathrm{eff}}}=\frac{1}{\mathrm{k}}\times \frac{1}{1-\frac{1}{2}}\\ \frac{1}{{\mathrm{k}}_{\mathrm{eff}}}=\frac{1}{\mathrm{k}}\times \frac{1}{\frac{2-1}{2}}\\ \frac{1}{{\mathrm{k}}_{\mathrm{eff}}}=\frac{2}{\mathrm{k}}\\ \mathrm{time}\mathrm{period}\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{{\mathrm{k}}_{\mathrm{eff}}}}\\ =2\mathrm{\pi }\sqrt{\frac{2\mathrm{m}}{\mathrm{k}}}\\ \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{pm}}{\mathrm{k}}}\\ \mathrm{p}=2\end{array}$