If ` (9 + 4 sqrt(5))^(n) = I + f `, `n` and `I` being positive integers and `f` is a proper fraction, show that ` (I-1 ) f + f^(2)` is an even integer.

Explanation

To solve the problem, we need to show that \( (I - 1)f + f^2 \) is an even integer given that \( (9 + 4\sqrt{5})^n = I + f \), where \( n \) and \( I \) are positive integers and \( f \) is a proper fraction.

### Step-by-Step Solution:

1. **Define \( f_1 \)**:
Let \( f_1 = (9 - 4\sqrt{5})^n \). This is the conjugate of \( (9 + 4\sqrt{5})^n \).

2. **Combine \( I + f \) and \( f_1 \)**:
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