Given the arithmetic progression with terms \(a, 2a, a^2\), the common difference \(d\) is:

\(d = 2a - a = a\)

\(d = a^2 - 2a\)

Equating the two expressions for \(d\):

\(a = a^2 - 2a\)

\(a^2 - 3a = 0\)

\(a(a - 3) = 0\)

Since \(a\) is positive, \(a = 3\).

Thus, \(d = 3\).

The sum of the first 50 terms \(S_{50}\) is given by:

\(S_{50} = \frac{50}{2} (2a + 49d)\)

Substitute \(a = 3\) and \(d = 3\):

\(S_{50} = 25 (2 \times 3 + 49 \times 3)\)

\(S_{50} = 25 (6 + 147)\)

\(S_{50} = 25 \times 153\)

\(S_{50} = 3825\)