Let the first term of the geometric progression (GP) be \( a \) and the common ratio be \( r \). The sum of the first three terms is: a + ar + ar\(^2\) = a(1 + r + r\(^2\)) \(S_3 = a(1 + r + r^2)\) The sum to infinity is: \(S_{\infty} = \frac{a}{1 - r}\) But: \(S_3 = \frac{1}{2} S_{\infty}\) Substituting the expressions gives: \(a(1 + r + r^2) = \frac{1}{2} \cdot \frac{a}{1 - r}\) Dividing by \( a \): \(1 + r + r^2 = \frac{1}{2(1 - r)}\) Cross-multiplying: \(2(1 + r + r^2)(1 - r) = 1\) Expanding and simplifying leads to: \(2(1 - r^3) = 1\) Thus: \(r^3 = \frac{1}{2}\) Taking the cube root: \(r = \sqrt[3]{\frac{1}{2}} = \frac{1}{\sqrt[3]{2}}\) Therefore, the positive common ratio of the progression is: \(\frac{1}{\sqrt[3]{2}}\).