Problem 149
Question
In a geometric progression consisting of positive terms, each term equals the sum of the next two terms. Then the common ratio of this progression equals (A) \(\frac{1}{2}(1-\sqrt{5})\) (B) \(\frac{1}{2} \sqrt{5}\) (C) \(\sqrt{5}\) (D) \(\frac{1}{2}(\sqrt{5}-1)\)
Step-by-Step Solution
Verified Answer
The common ratio is \( \frac{1}{2}(\sqrt{5} - 1) \), which is option (D).
1Step 1: Recognize the Problem Statement
We are given a geometric progression where each term equals the sum of the next two terms. We need to find the common ratio.
2Step 2: Define the Terms of Progression
Let the first term of the progression be \( a \) and the common ratio be \( r \). Thus, the terms of the progression are \( a, ar, ar^2, ar^3, \ldots \)
3Step 3: Apply the Given Condition
According to the problem, each term equals the sum of the next two terms. Therefore, \( a = ar + ar^2 \). Simplifying gives \( 1 = r + r^2 \).
4Step 4: Solve the Equation for r
Rearrange the equation to: \( r^2 + r - 1 = 0 \). This is a quadratic equation.
5Step 5: Apply the Quadratic Formula
For a quadratic equation \( ax^2 + bx + c = 0 \), the roots are \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Here, \( a = 1 \), \( b = 1 \), \( c = -1 \). The formula becomes \( r = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot -1}}{2 \cdot 1} \).
6Step 6: Simplify the Root Calculation
Calculate the discriminant: \( 1 + 4 = 5 \). Thus, \( r = \frac{-1 \pm \sqrt{5}}{2} \).
7Step 7: Determine the Common Ratio Validity
The roots are \( \frac{-1 + \sqrt{5}}{2} \) and \( \frac{-1 - \sqrt{5}}{2} \). Since the common ratio must be positive, choose \( \frac{-1 + \sqrt{5}}{2} \ > 0 \).
8Step 8: Compare with Given Options
The positive root is \( \frac{1}{2}(\sqrt{5} - 1) \), which corresponds to option (D).
Key Concepts
Quadratic Equations in Geometric ProgressionCommon Ratio ClarificationUnderstanding Positive Terms
Quadratic Equations in Geometric Progression
In mathematics, a quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable being solved for. Quadratic equations may have real, complex, or repeated solutions, all determined by the discriminant \( b^2 - 4ac \).
In the geometric progression problem, we encountered the equation \( r^2 + r - 1 = 0 \). Here, this is derived from the relationship between terms, each of which equals the sum of the two succeeding terms. We simplified the relationship between terms to create this quadratic form.
To solve for \( r \), the common ratio, we employed the quadratic formula:
In the geometric progression problem, we encountered the equation \( r^2 + r - 1 = 0 \). Here, this is derived from the relationship between terms, each of which equals the sum of the two succeeding terms. We simplified the relationship between terms to create this quadratic form.
To solve for \( r \), the common ratio, we employed the quadratic formula:
- \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Substituting the values, we have \( a = 1 \), \( b = 1 \), \( c = -1 \).
Common Ratio Clarification
The common ratio \( r \) is a crucial characteristic in a geometric progression, as it dictates the relationship between successive terms.
In our specific problem involving positive terms, the ratio must satisfy two conditions: it should maintain the consistency of the sequence, and importantly, it should ensure that each term equals the sum of the two following terms.
Mathematically, this means we rearrange our equation \( a = ar + ar^2 \) to \( 1 = r + r^2 \) as seen in the steps. This helps capture the essence of the sequence's behavior numerically when finding \( r \).
By solving this equation, we determine possible values for \( r \), which are then analyzed to choose the correct, meaningful root. The sequence indeed maintains its geometric nature through this derived positive root.
In our specific problem involving positive terms, the ratio must satisfy two conditions: it should maintain the consistency of the sequence, and importantly, it should ensure that each term equals the sum of the two following terms.
Mathematically, this means we rearrange our equation \( a = ar + ar^2 \) to \( 1 = r + r^2 \) as seen in the steps. This helps capture the essence of the sequence's behavior numerically when finding \( r \).
By solving this equation, we determine possible values for \( r \), which are then analyzed to choose the correct, meaningful root. The sequence indeed maintains its geometric nature through this derived positive root.
Understanding Positive Terms
In a geometric progression characterized by positive terms, all terms must be greater than zero for the series to hold true under the given specific conditions.
The exercise gives prominence to this by enforcing that our final chosen common ratio \( r \) needs to be positive. This ensures that the terms stay positive as the progression advances. It avoids alternations into negative or zero values, which would break the pattern of the sequence.
This condition is particularly handled during the selection of the roots derived from the quadratic solution. Out of the roots obtained, we focus on the one that meets the necessary condition:
The exercise gives prominence to this by enforcing that our final chosen common ratio \( r \) needs to be positive. This ensures that the terms stay positive as the progression advances. It avoids alternations into negative or zero values, which would break the pattern of the sequence.
This condition is particularly handled during the selection of the roots derived from the quadratic solution. Out of the roots obtained, we focus on the one that meets the necessary condition:
- \( \frac{-1 + \sqrt{5}}{2} > 0 \)
Other exercises in this chapter
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