Find Index of Smallest Triangular Number

A triangular number is a number that can be represented as a triangle with dots. The nth triangular number is given by the formula:

\[ T_n = \frac{n(n+1)}{2} \]

Write a Python function `find_index(n)` that finds the index of the smallest triangular number with exactly `n` digits.

#### Example Usage
```python [main.nopy]
find_index(2)  # Output: 4
find_index(3)  # Output: 14
find_index(4)  # Output: 45
```

#### Constraints
- The input `n` will be a positive integer.
- The function should return the smallest index `k` such that the triangular number \( T_k \) has exactly `n` digits.

import math

def find_index(n):
    """
    Find the index of the smallest triangular number with exactly n digits.

    Args:
        n (int): The number of digits.

    Returns:
        int: The index of the smallest triangular number with n digits.
    """
    # Placeholder for the solution
    pass

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Intermediate math functions

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