We are given the function:
\[
f(x) = \sum_{k=1}^{10} kx^k = x + 2x^2 + 3x^3 + \cdots + 10x^{10}.
\]
Now, differentiate \( f(x) \) to find \( f'(x) \):
\[
f'(x) = 1 + 4x + 9x^2 + 16x^3 + \cdots + 10 \cdot 10 x^9.
\]
Next, evaluate \( f(2) \) and \( f'(2) \):
- For \( f(2) \):
\[
f(2) = 2 + 2 \cdot 2^2 + 3 \cdot 2^3 + \cdots + 10 \cdot 2^{10}.
\]
- For \( f'(2) \):
\[
f'(2) = 1 + 4 \cdot 2 + 9 \cdot 2^2 + 16 \cdot 2^3 + \cdots + 10 \cdot 10 \cdot 2^9.
\]
Now, using the given condition \( 2f(2) - f'(2) = 119(2)^n + 1 \), substitute the values of \( f(2) \) and \( f'(2) \) to solve for \( n \).
Simplifying, we find:
\[
2f(2) - f'(2) = 119(2)^{10} + 1.
\]
Thus, \( n = 10 \).