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 [2007](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 corresponds to option (D).
1Step 1: Define Geometric Progression
A geometric progression is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. Let's denote the first term by \( a \) and the common ratio by \( r \). Then, the terms of the geometric progression would be \( a, ar, ar^2, ar^3, \ldots \).
2Step 2: Set Up the Equation
According to the problem, each term is equal to the sum of the next two terms. Taking the first three terms, we have \( a = ar + ar^2 \). This is our main equation.
3Step 3: Simplify the Equation
Divide the entire equation \( a = ar + ar^2 \) by \( a \) to simplify:\[1 = r + r^2.\]Now, solve this equation for \( r \).
4Step 4: Solve the Quadratic Equation
The simplified quadratic equation is \( r^2 + r - 1 = 0 \). Solve it using the quadratic formula:\[r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]where \( a = 1 \), \( b = 1 \), and \( c = -1 \).
5Step 5: Apply the Quadratic Formula
Substitute the coefficients into the quadratic formula:\[r = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2}.\]
6Step 6: Identify Valid Solutions
We need the solution that results in a positive common ratio because the sequence consists of positive terms. The positive solution is:\[r = \frac{-1 + \sqrt{5}}{2}.\]
7Step 7: Verify Against Options
Compare the solution \( \frac{-1 + \sqrt{5}}{2} \) with the options provided. This matches option \( \frac{1}{2}(\sqrt{5} - 1) \) since \( \frac{-1 + \sqrt{5}}{2} = \frac{1}{2}(\sqrt{5} - 1) \).
Key Concepts
Quadratic EquationCommon RatioSequence
Quadratic Equation
A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \).
It is particularly significant in algebra because it contains a squared term. To solve quadratic equations, one common method is the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula offers the solutions to the equation by providing the possible values of \( x \) that satisfy it. Let's break down the formula:
- **\( -b \)**: The opposite of the coefficient of the linear term.
- **Discriminant, \( b^2 - 4ac \)**: Determines the nature of the roots.
If the discriminant is positive, the equation has two real and distinct roots. If it is zero, the equation has one real and repeated root. If negative, the roots are complex.
- **\( 2a \)**: A divisor that influences the scaling of the equation's solutions.In the context of the given problem, after setting up the equation \( 1 = r + r^2 \), rearranging gives us the quadratic \( r^2 + r - 1 = 0 \). By applying the quadratic formula, we can solve for \( r \) to find the common ratio.
It is particularly significant in algebra because it contains a squared term. To solve quadratic equations, one common method is the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula offers the solutions to the equation by providing the possible values of \( x \) that satisfy it. Let's break down the formula:
- **\( -b \)**: The opposite of the coefficient of the linear term.
- **Discriminant, \( b^2 - 4ac \)**: Determines the nature of the roots.
If the discriminant is positive, the equation has two real and distinct roots. If it is zero, the equation has one real and repeated root. If negative, the roots are complex.
- **\( 2a \)**: A divisor that influences the scaling of the equation's solutions.In the context of the given problem, after setting up the equation \( 1 = r + r^2 \), rearranging gives us the quadratic \( r^2 + r - 1 = 0 \). By applying the quadratic formula, we can solve for \( r \) to find the common ratio.
Common Ratio
The common ratio in a geometric sequence is the constant factor by which each term in the sequence is multiplied to obtain the next term. In a general geometric progression defined by terms \( a, ar, ar^2, ar^3, \ldots \), the common ratio \( r \) is crucial for understanding how the sequence progresses.
In the original problem, we were tasked with determining the common ratio such that each term equals the sum of the next two terms. We set the equation \( a = ar + ar^2 \), leading to \( 1 = r + r^2 \). Solving this equation provides us with the possible values of \( r \) that make the sequence behave as described.
By solving the quadratic equation \( r^2 + r - 1 = 0 \), we derived two possible values for \( r \):
- The solutions \( r = \frac{-1 \pm \sqrt{5}}{2} \) result in the common ratio options provided in the exercise.
- Since all terms are positive, we choose \( r = \frac{-1 + \sqrt{5}}{2} \), known as the golden ratio conjugate, which ensures each term is indeed the sum of its following two positive terms.
In the original problem, we were tasked with determining the common ratio such that each term equals the sum of the next two terms. We set the equation \( a = ar + ar^2 \), leading to \( 1 = r + r^2 \). Solving this equation provides us with the possible values of \( r \) that make the sequence behave as described.
By solving the quadratic equation \( r^2 + r - 1 = 0 \), we derived two possible values for \( r \):
- The solutions \( r = \frac{-1 \pm \sqrt{5}}{2} \) result in the common ratio options provided in the exercise.
- Since all terms are positive, we choose \( r = \frac{-1 + \sqrt{5}}{2} \), known as the golden ratio conjugate, which ensures each term is indeed the sum of its following two positive terms.
Sequence
In mathematics, a sequence refers to an ordered list of numbers following a particular pattern or rule. Sequences can be categorized into various types, with geometric progressions being one such type characterized by a constant ratio between consecutive terms.
A geometric sequence, specifically, has each term after the first obtained by multiplying the previous term by a fixed, non-zero number called the common ratio \( r \). An example of this type of sequence would be \( a, ar, ar^2, ar^3, \ldots \).
Sequences can be used to model and solve real-world problems in various fields, from finance to physics, analyzing repetitive patterns over time.
In our problem, understanding the sequence pattern is crucial because it directly affects how we set up and solve the equation involving the common ratio. Recognizing that each term equals the sum of the next two terms allows us to formulate the initial relationship \( a = ar + ar^2 \), leading to the discovery of the required common ratio to maintain the pattern across the sequence.
The structure of sequences, especially geometric progressions, provides a strong foundation for solving more complex mathematical problems by defining precise relationships between their terms.
A geometric sequence, specifically, has each term after the first obtained by multiplying the previous term by a fixed, non-zero number called the common ratio \( r \). An example of this type of sequence would be \( a, ar, ar^2, ar^3, \ldots \).
Sequences can be used to model and solve real-world problems in various fields, from finance to physics, analyzing repetitive patterns over time.
In our problem, understanding the sequence pattern is crucial because it directly affects how we set up and solve the equation involving the common ratio. Recognizing that each term equals the sum of the next two terms allows us to formulate the initial relationship \( a = ar + ar^2 \), leading to the discovery of the required common ratio to maintain the pattern across the sequence.
The structure of sequences, especially geometric progressions, provides a strong foundation for solving more complex mathematical problems by defining precise relationships between their terms.
Other exercises in this chapter
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