2025 IGCSE Additional Mathematics May/June 0606/22 Q7

The first three terms of an arithmetic progression can be written as

\(2\ln(x^3),\quad5\ln(x^2),\quad2\ln(x^7).\)

1. Given that \(x>1\), find the least number of terms for the sum of this progression to be greater than \(43\ln(x^{24})\).[6]
2. Given that the 25th term of this progression is equal to \(408\), find the exact value of \(x\).[3]