If each term of a geometric progression (GP) is positive and is the s…

The correct answer is Option (3) → $\frac{\sqrt{5}+1}{2}$

Step 1: Let the GP be $a, ar, ar^2, ar^3, \dots$

The property given:

$\text{Each term is the sum of the two preceding terms: } ar^n = ar^{n-1} + ar^{n-2}, \ n \ge 2$

Divide both sides by $ar^{n-2}$ (positive, non-zero):

$r^2 = r + 1$

Step 2: Solve quadratic equation

$r^2 - r - 1 = 0$

$r = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}$

Since all terms are positive, we take the positive root:

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