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 serieseffective spring constant is Keffective=Keff is1Keff=1K+12K+14K+−−−1Keff=1K1+12+122+... , the paranthesis part is in geometric progression whose first term =1 and common ratio is r=12sum to infinite terms is S∞=a1-r where a is first term, r is common ratio  1Keff=1Ka1-r=1K11-121Keff=1K21Keff=K2Frequency   η=12πKeffm=12πK2m due to series of springs new frequency is=12πK2m