Base case: For \(n = 1\), \(2025^1 + 47^1 - 2 = 2070 = 45 \times 46\), which is divisible by 46.

Inductive step: Assume \(2025^k + 47^k - 2\) is divisible by 46 for some positive integer \(k\).

Then, consider \(2025^{k+1} + 47^{k+1} - 2\).

\(2025^{k+1} + 47^{k+1} - 2 = (2024 + 1)2025^k + (46 + 1)47^k - 2\)

This can be rewritten as:

\(2024 \times 2025^k + 46 \times 47^k + 2025^k + 47^k - 2\)

\(= 46(44 \times 2025^k + 47^k) + (2025^k + 47^k - 2)\)

By the inductive hypothesis, \(2025^k + 47^k - 2\) is divisible by 46.

Hence, \(2025^{k+1} + 47^{k+1} - 2\) is divisible by 46.

Therefore, by induction, the statement is true for every positive integer \(n\).