Question

In a GP consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of the GP is:
1. \(\dfrac{1-\sqrt{5}}{2}\)
2. \(\dfrac{\sqrt{5}}{2}\)
3. \(\sqrt{5}\)
4. \(\dfrac{\sqrt{5}-1}{2}\)

Answer 1

Correct Answer - Option 4 : \(\dfrac{\sqrt{5}-1}{2}\)

Concept:

Geometric Progression (GP):

• The series of numbers where the ratio of any two consecutive terms is the same, is called a Geometric Progression.

• A Geometric Progression of n terms, with first term a and common ratio r, is represented as:

  a, ar, ar2, ar3, ..., arn-2, arn-1.

 

The solution to the quadratic equation ax² + bx + c = 0 is given by: x = \(\rm \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\).

 

Calculation:

Let's say that the GP is a, ar, ar2, ar3, ..., arn-2, arn-1, with first term a and common ratio r.

Since the GP has only positive terms, it means that a > 0 and r > 0.

According to the question, arⁿ = arⁿ⁺¹ + arn+2, for any n. Dividing this equation by arⁿ (which is > 0), we get:

⇒ 1 = r + r²

⇒ r² + r - 1 = 0

⇒ r = \(\rm \dfrac{-1 \pm \sqrt{1^2-4(1)(-1)}}{2}\)

⇒ r = \(\rm \dfrac{-1 + \sqrt{5}}{2}\) OR r = \(\rm \dfrac{-1 - \sqrt{5}}{2}\).

Since, r is not negative (a > 0 and r > 0), the common ratio of the series is r = \(\rm \dfrac{-1 + \sqrt{5}}{2}\).