In a geometric progression consisting of positive terms, each term equals the sum of the next two te

Q: Wait, why can we divide by $a$?

Great question! We are given that the geometric progression consists of positive terms. This implies the first term $a$ must be greater than zero. Since $a \neq 0$, dividing by $a$ is perfectly valid and helps us isolate the common ratio $r$.

Q: Why do we reject the negative root?

The problem states that all terms in the progression are positive. If we chose the negative root $r = \frac{-1 - \sqrt{5}}{2}$, the terms would alternate in sign (positive, negative, positive, negative), which violates the problem's condition. Thus, we must discard it.

Q: Is there a connection to the Golden Ratio?

Absolutely! The value $\frac{\sqrt{5}-1}{2}$ is the reciprocal of the Golden Ratio $\phi = \frac{1+\sqrt{5}}{2}$. It is a beautiful example of how geometric progressions can naturally lead to fundamental mathematical constants.

Question

In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals

Options:

(A) $\sqrt{5}$

(B) $\frac{1}{2}(\sqrt{5} - 1)$

(C) $\frac{1}{2}(1 - \sqrt{5})$

(D) $\frac{\sqrt{5}}{2}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the threshold of a sequence that grows with a rhythm so precise, it mirrors the very patterns found in nature. We are dealing with a Geometric Progression (G.P.), a sequence where each term is born from the previous one by a constant multiplier, the common ratio $r$.

Let us define our terms as $a, ar, ar^2, ar^3, \dots$. The problem provides a fascinating constraint: each term is the sum of the next two.

This is not just an algebraic puzzle; it is a structural rule that defines the DNA of this sequence. We start by writing this condition for the first term:

$$a = ar + ar^2$$

This equation is our anchor. It tells us that the first term is not independent; it is bound to the ratio $r$ in a specific, rigid way.

### The Algebraic Dance

Now, we face a moment of simplification. We have $a = ar + ar^2$. Since we know all terms are positive, we are guaranteed that $a > 0$.

This is our green light to divide the entire equation by $a$. By doing so, we strip away the specific starting value and reveal the pure relationship between the terms:

$$1 = r + r^2$$

This is the heart of the problem. We have transformed a sequence problem into a quadratic equation:

$$r^2 + r - 1 = 0$$

This is a classic quadratic form $Ar^2 + Br + C = 0$, where $A=1$, $B=1$, and $C=-1$.

### The Quadratic Crossroads

To find $r$, we invoke the quadratic formula:

$$r = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$$

Substituting our coefficients, we get:

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

Simplifying the discriminant, we find $1 + 4 = 5$. Thus, our potential ratios are:

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

We are left with two candidates: $r_1 = \frac{-1 + \sqrt{5}}{2}$ and $r_2 = \frac{-1 - \sqrt{5}}{2}$.

### The Final Filter

In mathematics, as in life, we must filter our results against the reality of our constraints. We were told the terms are positive.

If $r$ were negative, the terms would oscillate between positive and negative values. Therefore, $r$ must be positive.

Looking at our two roots, $r_2 = \frac{-1 - \sqrt{5}}{2}$ is clearly negative, so we must reject it. This leaves us with the only valid solution:

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

This result is not just a number; it is the reciprocal of the Golden Ratio, a constant that appears in everything from the spiral of galaxies to the arrangement of leaves. You have successfully navigated the algebra to uncover a fundamental geometric truth.

Animated Solution for Mathematics - Sequence and Series: In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of its progression is equals

Select Answer:

Visualized Solution

Defining the Geometric Progression

Applying the Given Condition

Simplifying the Equation

Forming the Quadratic Equation

The Quadratic Formula

Substituting the Coefficients

Calculating the Roots

Filtering the Valid Common Ratio

Final Answer

The Sigma Insight: Geometric Progression (G.P.)

Analyzing the Setup

The Algebraic Dance

The Quadratic Crossroads

The Final Filter

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