2025 AHS PRELIMS P1 Q15

A new type of energy harvesting device is being developed to capture energy from small environmental vibrations. The engineers observe that the fraction of energy captured by the device during the \(n\)th second of continuous operation, relative to its maximum capacity, follows a specific pattern.

1. Write down the fraction of energy captured for the \(n\)th second of operation.[1]
2. The team needs to find the total energy captured after \(K\) seconds of continuous operation. Show that the total energy captured after \(K\) seconds is \(\dfrac12\left(1-\dfrac{1}{K+1}\right)\).[3]
3. The energy harvesting device is considered a success if it is able to harvest at least 49.8% of its maximum capacity. Find the minimum number of seconds for which the device is considered a success.[3]