For a geometric progression, the common ratio \(r\) is given by:

\(r = \frac{\text{second term}}{\text{first term}} = \frac{\frac{a^2}{a+2}}{a} = \frac{a}{a+2}\)

The sum to infinity \(S_\infty\) of a geometric series is given by:

\(S_\infty = \frac{a}{1 - r}\)

Substitute \(r = \frac{a}{a+2}\) and \(S_\infty = 264\):

\(\frac{a}{1 - \frac{a}{a+2}} = 264\)

Simplify the denominator:

\(1 - \frac{a}{a+2} = \frac{a+2-a}{a+2} = \frac{2}{a+2}\)

Thus, the equation becomes:

\(\frac{a}{\frac{2}{a+2}} = 264\)

Multiply both sides by \(\frac{2}{a+2}\):

\(a(a+2) = 264 \times 2\)

\(a(a+2) = 528\)

\(a^2 + 2a - 528 = 0\)

Factorize the quadratic equation:

\((a - 22)(a + 24) = 0\)

Thus, \(a = 22\) or \(a = -24\). Since \(a\) is positive, \(a = 22\).