If each term of a geometric progression (GP) is positive and is the sum of two preceding terms, then the common ratio of the GP is:

Let the first term of the GP be $a$ and the common ratio be $r$, where $a > 0$.

The GP is: $a, ar, ar^2, ar^3, ar^4, ...$

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Each term is the sum of the two preceding terms. Taking the third term:

$ar^2 = a + ar$

Dividing by $a$ (since $a > 0$):

$r^2 = 1 + r$

$r^2 - r - 1 = 0$

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Using the quadratic formula:

$r = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}$

$r = \frac{1 \pm \sqrt{1 + 4}}{2}$

$r = \frac{1 \pm \sqrt{5}}{2}$

This gives two values:

$r = \frac{1 + \sqrt{5}}{2}$ or $r = \frac{1 - \sqrt{5}}{2}$

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Since all terms of the GP are positive, the common ratio must be positive.

$\frac{1 - \sqrt{5}}{2} < 0$ (since $\sqrt{5} > 2$)

$\frac{1 + \sqrt{5}}{2} > 0$

Therefore, the common ratio is $r = \frac{1 + \sqrt{5}}{2}$.