Question

Every term of a geometric progression is positive, and every term is the sum of the two preceding terms. Then the common ratio of the geometric progression is:

• A. \(1\)
• B. \(\dfrac{\sqrt5-1}{2}\)
• C. \(\dfrac{1-\sqrt5}{2}\)
• D. \(\dfrac{1+\sqrt5}{2}\)

Hint:

When a GP satisfies a recursive relation, express consecutive terms using powers of the common ratio and derive an equation in \(r\).

Answer 1

The Correct Option is D

Concept: In a geometric progression, \[ a, ar, ar^2, ar^3,\ldots \] If every term equals the sum of the previous two terms, then the common ratio satisfies a quadratic equation.

Step 1: Use the GP structure. Suppose three consecutive terms are: \[ a,\ ar,\ ar^2 \] Given condition: \[ ar^2=ar+a \]

Step 2: Simplify the equation. Divide both sides by \(a\): \[ r^2=r+1 \] Rearranging, \[ r^2-r-1=0 \]

Step 3: Solve the quadratic equation. Using quadratic formula: \[ r=\frac{1\pm\sqrt{1+4}}{2} \] \[ r=\frac{1\pm\sqrt5}{2} \] Since every term of the GP is positive, the common ratio must be positive. Therefore, \[ r=\frac{1+\sqrt5}{2} \] Hence, \[ \boxed{\frac{1+\sqrt5}{2}} \]