Not identifying the common ratio

Oct 13, 2025 by magicalceylontours.com 74 views

Does the List of Numbers 3/4, 3/8, 3/16 Form a Geometric Progression (G.P.)?

A geometric progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this article, we will explore whether the list of numbers 3/4, 3/8, 3/16 forms a geometric progression.

A geometric progression is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio. The general form of a geometric progression is:

a, ar, ar^2, ar^3, ...

where 'a' is the first term and 'r' is the common ratio.

To determine whether the list of numbers 3/4, 3/8, 3/16 forms a geometric progression, we need to find the common ratio between the terms.

Let's start by finding the ratio between the second term (3/8) and the first term (3/4):

(3/8) / (3/4) = (3/8) × (4/3) = 1/2

Now, let's find the ratio between the third term (3/16) and the second term (3/8):

(3/16) / (3/8) = (3/16) × (8/3) = 1/2

As we can see, the ratio between the second term and the first term is the same as the ratio between the third term and the second term, which is 1/2. This means that the list of numbers 3/4, 3/8, 3/16 has a common ratio of 1/2.

Based on the calculations above, we can conclude that the list of numbers 3/4, 3/8, 3/16 does form a geometric progression with a common ratio of 1/2.

Understanding geometric progression is important in many areas of mathematics, science, and engineering. For example, geometric progression is used to model population growth, financial investments, and electrical circuits.

Geometric progression has many real-world applications, including:

Population growth: Geometric progression can be used to model population growth, where the population at each time period is obtained by multiplying the previous population by a fixed number.
Financial investments: Geometric progression can be used to calculate the future value of an investment, where the interest rate is applied to the previous balance to obtain the next balance.
Electrical circuits: Geometric progression can be used to model the behavior of electrical circuits, where the voltage at each node is obtained by multiplying the previous voltage by a fixed number.

Here are some tips for solving geometric progression problems:

Identify the common ratio: The common ratio is the key to solving geometric progression problems. Make sure to identify the common ratio between the terms.
Use the formula: The formula for the nth term of a geometric progression is an = a1 × r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.
Practice, practice, practice: Geometric progression problems can be challenging, so make sure to practice solving them to build your skills.

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 term by a fixed, non-zero number called the common ratio.

Q: What is the common ratio in a geometric progression?

A: The common ratio is the fixed number that is multiplied by each term to obtain the next term in the sequence.

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

A: To find the common ratio, divide each term by the previous term. If the result is the same for each pair of terms, then the common ratio has been found.

Q: What is the formula for the nth term of a geometric progression?

A: The formula for the nth term of a geometric progression is an = a1 × r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.

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

A: To determine if a sequence is a geometric progression, check if each term is obtained by multiplying the previous term by a fixed number. If this is the case, then the sequence is a geometric progression.

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

A: Geometric progression has many real-world applications, including:

Population growth: Geometric progression can be used to model population growth, where the population at each time period is obtained by multiplying the previous population by a fixed number.
Financial investments: Geometric progression can be used to calculate the future value of an investment, where the interest rate is applied to the previous balance to obtain the next balance.
Electrical circuits: Geometric progression can be used to model the behavior of electrical circuits, where the voltage at each node is obtained by multiplying the previous voltage by a fixed number.

Q: How do I solve geometric progression problems?

A: To solve geometric progression problems, follow these steps:

Identify the common ratio: The common ratio is the key to solving geometric progression problems. Make sure to identify the common ratio between the terms.
Use the formula: The formula for the nth term of a geometric progression is an = a1 × r^(n-1), where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.
Practice, practice, practice: Geometric progression problems can be challenging, so make sure to practice solving them to build your skills.

Q: What are some common mistakes to avoid when working with geometric progression?

A: Some common mistakes to avoid when working with geometric progression include:

Not identifying the common ratio: Make sure to identify the common ratio between the terms.
Not using the formula: Use the formula for the nth term of a geometric progression to solve problems.
Not practicing: Geometric progression problems can be challenging, so make sure to practice solving them to build your skills.

Q: How do I determine the sum of a geometric progression?

A: To determine the sum of a geometric progression, use the formula:

S = a1 × (1 - r^n) / (1 - r)

where S is the sum, a1 is the first term, r is the common ratio, and n is the number of terms.

Q: What are some advanced topics in geometric progression?

A: Some advanced topics in geometric progression include:

Infinite geometric progressions: An infinite geometric progression is a geometric progression with an infinite number of terms.
Convergence of geometric progressions: A geometric progression converges if the absolute value of the common ratio is less than 1.
Divergence of geometric progressions: A geometric progression diverges if the absolute value of the common ratio is greater than 1.

In conclusion, geometric progression is a fundamental concept in mathematics that has many real-world applications. By understanding the basics of geometric progression, you can solve problems with confidence and apply the concept to real-world situations.