If $k+1, 2k-1, 3k+1$ Are Three Consecutive Terms Of A Geometric Progression, Find The Possible Values Of The Common Ratio.

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Introduction

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In this problem, we are given three consecutive terms of a geometric progression: $k+1, 2k-1, 3k+1$. Our goal is to find the possible values of the common ratio.

Understanding the Problem

To start, let's recall the definition of a geometric progression. If $a, ar, ar^2, ...$ is a geometric progression with first term $a$ and common ratio $r$, then each term is obtained by multiplying the previous one by $r$. In our problem, we have three consecutive terms: $k+1, 2k-1, 3k+1$. We need to find the common ratio that connects these terms.

Setting Up the Equation

Since the three terms are consecutive in a geometric progression, we can write the following equation:

2k−1k+1=3k+12k−1\frac{2k-1}{k+1} = \frac{3k+1}{2k-1}

This equation represents the relationship between the three terms, where the ratio of the second term to the first term is equal to the ratio of the third term to the second term.

Solving the Equation

To solve for the common ratio, we can start by cross-multiplying the equation:

(2k−1)2=(k+1)(3k+1)(2k-1)^2 = (k+1)(3k+1)

Expanding both sides of the equation, we get:

4k2−4k+1=3k2+4k+14k^2 - 4k + 1 = 3k^2 + 4k + 1

Simplifying the equation, we get:

k2−8k=0k^2 - 8k = 0

Factoring out $k$, we get:

k(k−8)=0k(k - 8) = 0

This gives us two possible values for $k$: $k = 0$ or $k = 8$.

Finding the Common Ratio

Now that we have found the possible values of $k$, we can substitute them back into the original equation to find the common ratio.

For $k = 0$, we have:

2(0)−10+1=3(0)+12(0)−1\frac{2(0)-1}{0+1} = \frac{3(0)+1}{2(0)-1}

Simplifying the equation, we get:

−1=−1-1 = -1

This is a true statement, so $k = 0$ is a possible value.

For $k = 8$, we have:

2(8)−18+1=3(8)+12(8)−1\frac{2(8)-1}{8+1} = \frac{3(8)+1}{2(8)-1}

Simplifying the equation, we get:

159=2515\frac{15}{9} = \frac{25}{15}

This is also a true statement, so $k = 8$ is a possible value.

Conclusion

In this problem, we were given three consecutive terms of a geometric progression: $k+1, 2k-1, 3k+1$. We found the possible values of the common ratio by setting up an equation and solving for $k$. We found that $k = 0$ and $k = 8$ are the possible values, and we verified that these values satisfy the original equation. Therefore, the possible values of the common ratio are $r = -1$ and $r = \frac{5}{3}$.

Final Answer

The final answer is: −1,53\boxed{-1, \frac{5}{3}}

Introduction

In our previous article, we explored the problem of finding the possible values of the common ratio for a geometric progression with three consecutive terms: $k+1, 2k-1, 3k+1$. We derived the equation $k^2 - 8k = 0$ and found that $k = 0$ and $k = 8$ are the possible values. In this article, we will provide a Q&A section to further clarify the solution and answer any additional questions.

Q: What is a geometric progression?

A: A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Q: How do we find the common ratio in a geometric progression?

A: To find the common ratio, we can use the formula $r = \frac{a_{n+1}}{a_n}$, where $a_n$ is the nth term and $a_{n+1}$ is the (n+1)th term.

Q: What is the relationship between the three terms in the problem?

A: The three terms are consecutive in a geometric progression, so we can write the equation $\frac{2k-1}{k+1} = \frac{3k+1}{2k-1}$.

Q: How do we solve for the common ratio?

A: We can start by cross-multiplying the equation and simplifying it to get $k^2 - 8k = 0$.

Q: What are the possible values of $k$?

A: The possible values of $k$ are $k = 0$ and $k = 8$.

Q: How do we find the common ratio for each possible value of $k$?

A: We can substitute each possible value of $k$ back into the original equation to find the common ratio.

Q: What are the possible values of the common ratio?

A: The possible values of the common ratio are $r = -1$ and $r = \frac{5}{3}$.

Q: Can you provide an example of a geometric progression with a common ratio of $r = -1$?

A: Yes, an example of a geometric progression with a common ratio of $r = -1$ is $1, -1, 1, -1, ...$.

Q: Can you provide an example of a geometric progression with a common ratio of $r = \frac{5}{3}$?

A: Yes, an example of a geometric progression with a common ratio of $r = \frac{5}{3}$ is $3, 5, 25/3, 125/9, ...$.

Q: How do we determine if a sequence is a geometric progression?

A: To determine if a sequence is a geometric progression, we can check if the ratio of each term to the previous term is constant.

Q: What are some real-world applications of geometric progressions?

A: Geometric progressions have many real-world applications, including finance, population growth, and electrical engineering.

Conclusion

In this Q&A article, we provided additional information and clarification on the problem of finding the possible values of the common ratio for a geometric progression with three consecutive terms: $k+1, 2k-1, 3k+1$. We answered questions on the definition of a geometric progression, how to find the common ratio, and provided examples of geometric progressions with common ratios of $r = -1$ and $r = \frac{5}{3}$. We also discussed how to determine if a sequence is a geometric progression and provided information on real-world applications of geometric progressions.

Final Answer

The final answer is: −1,53\boxed{-1, \frac{5}{3}}