The energy added to a humanoid robot's backup cell increases by a constant number of units each hour of its operation. The energy added during the 8th hour is 26 units, and the total cumulative energy stored over the first 12 hours is 258 units.
Find the energy added during the 1st hour and determine the minimum number of full hours the robot needs to operate before its cumulative energy stored exceeds 1000 units.[5]
The robot requires compute tokens to run its onboard AI models. The robot's server wallet starts off with 10 000 tokens. At the beginning of each month starting from the first month, a fixed number of \(K\) tokens is added to the wallet. At the end of each month, the total number of tokens in the wallet increases by 2% due to network yields. Immediately after, the robot consumes exactly 2500 tokens for a monthly software update.
Let \(u_n\) denote the number of tokens in the wallet at the end of \(n\)th month after the monthly software update.
Write down the expressions for \(u_1\) and \(u_2\) in terms of \(K\).[2]
Hence show that \(u_n=1.02^n(10\,000)+\alpha K(1.02^n-1)-125\,000(1.02^n-1)\), where \(\alpha\) is a constant to be determined.[3]
Find the minimum integer value of \(K\) to ensure the robot's wallet never runs out of tokens in the long run.[2]