The $3^{\text{rd}}$ Term And The $9^{\text{th}}$ Term Of A Geometric Progression (G.P.) Are 54 And 39,366, Respectively. Find The Common Ratio.

Introduction

A geometric progression (G.P.) 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. The general form of a G.P. is given by:

$$a, ar, ar^2, ar^3, \ldots$$

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

The $3^{\text{rd}}$ term and the $9^{\text{th}}$ term of a G.P.

Given that the $3^{\text{rd}}$ term of a G.P. is 54 and the $9^{\text{th}}$ term is 39,366, we can use this information to find the common ratio.

The $3^{\text{rd}}$ term of a G.P.

The $3^{\text{rd}}$ term of a G.P. is given by:

$$ar^2 = 54$$

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

The $9^{\text{th}}$ term of a G.P.

The $9^{\text{th}}$ term of a G.P. is given by:

$$ar^8 = 39366$$

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

Finding the common ratio

To find the common ratio, we can divide the $9^{\text{th}}$ term by the $3^{\text{rd}}$ term:

$$\frac{ar^8}{ar^2} = \frac{39366}{54}$$

Simplifying the expression, we get:

$$r^6 = 727$$

Taking the sixth root of both sides, we get:

$$r = \sqrt[6]{727}$$

Calculating the value of the common ratio

Using a calculator, we can calculate the value of the common ratio:

$$r = \sqrt[6]{727} \approx 2.5$$

Conclusion

In this article, we used the given information about the $3^{\text{rd}}$ term and the $9^{\text{th}}$ term of a geometric progression (G.P.) to find the common ratio. We first expressed the $3^{\text{rd}}$ term and the $9^{\text{th}}$ term in terms of the first term and the common ratio. We then divided the $9^{\text{th}}$ term by the $3^{\text{rd}}$ term to get an expression for the common ratio. Finally, we calculated the value of the common ratio using a calculator.

Example

Suppose we are given the $5^{\text{th}}$ term and the $11^{\text{th}}$ term of a G.P. as 243 and 59,049, respectively. We can use the same method to find the common ratio.

The $5^{\text{th}}$ term of a G.P.

The $5^{\text{th}}$ term of a G.P. is given by:

$$ar^4 = 243$$

The $11^{\text{th}}$ term of a G.P.

The $11^{\text{th}}$ term of a G.P. is given by:

$$ar^{10} = 59049$$

Dividing the $11^{\text{th}}$ term by the $5^{\text{th}}$ term, we get:

$$\frac{ar^{10}}{ar^4} = \frac{59049}{243}$$

Simplifying the expression, we get:

$$r^6 = 243$$

Taking the sixth root of both sides, we get:

$$r = \sqrt[6]{243}$$

Using a calculator, we can calculate the value of the common ratio:

$$r = \sqrt[6]{243} \approx 2.5$$

Applications of geometric progressions

Geometric progressions have many applications in mathematics, science, and engineering. Some examples include:

• Finance: Geometric progressions are used to calculate compound interest and investment returns.
• Physics: Geometric progressions are used to describe the motion of objects under constant acceleration.
• Engineering: Geometric progressions are used to design and analyze electrical circuits and mechanical systems.

Conclusion

In this article, we used the given information about the $3^{\text{rd}}$ term and the $9^{\text{th}}$ term of a geometric progression (G.P.) to find the common ratio. We first expressed the $3^{\text{rd}}$ term and the $9^{\text{th}}$ term in terms of the first term and the common ratio. We then divided the $9^{\text{th}}$ term by the $3^{\text{rd}}$ term to get an expression for the common ratio. Finally, we calculated the value of the common ratio using a calculator. Geometric progressions have many applications in mathematics, science, and engineering, and are an important concept in mathematics.

Introduction

Geometric progressions (G.P.) are a fundamental concept in mathematics, and are used to describe a wide range of phenomena in science, engineering, and finance. In this article, we will answer some common questions about geometric progressions, including how to find the common ratio, how to calculate the nth term, and how to use geometric progressions to solve real-world problems.

Q: What is a geometric progression?

A: A geometric progression (G.P.) 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 I find the common ratio of a geometric progression?

A: To find the common ratio of a geometric progression, you can use the formula:

$$r = \sqrt[n-1]{\frac{a_n}{a_1}}$$

where $a_n$ is the nth term of the progression, $a_1$ is the first term, and $n$ is the number of terms.

Q: How do I calculate the nth term of a geometric progression?

A: To calculate the nth term of a geometric progression, you can use the formula:

$$a_n = a_1 \cdot r^{n-1}$$

where $a_n$ is the nth term of the progression, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

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

A: The formula for the sum of a geometric progression is:

$$S_n = \frac{a_1(1-r^n)}{1-r}$$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Q: How do I use geometric progressions to solve real-world problems?

A: Geometric progressions can be used to solve a wide range of real-world problems, including:

• Finance: Geometric progressions can be used to calculate compound interest and investment returns.
• Physics: Geometric progressions can be used to describe the motion of objects under constant acceleration.
• Engineering: Geometric progressions can be used to design and analyze electrical circuits and mechanical systems.

Q: What are some common applications of geometric progressions?

A: Some common applications of geometric progressions include:

• Compound interest: Geometric progressions can be used to calculate compound interest and investment returns.
• Population growth: Geometric progressions can be used to model population growth and decline.
• Electrical circuits: Geometric progressions can be used to design and analyze electrical circuits.

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

A: To determine if a sequence is a geometric progression, you can check if each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

Q: What is the difference between a geometric progression and an arithmetic progression?

A: The main difference between a geometric progression and an arithmetic progression is that in a geometric progression, each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, while in an arithmetic progression, each term after the first is found by adding a fixed number called the common difference.

Q: Can I use geometric progressions to solve problems with negative numbers?

A: Yes, you can use geometric progressions to solve problems with negative numbers. However, you will need to take into account the sign of the common ratio and the terms of the progression.

Q: How do I use geometric progressions to solve problems with fractions?

A: You can use geometric progressions to solve problems with fractions by using the formula for the nth term of a geometric progression and substituting the fraction for the common ratio.

Q: Can I use geometric progressions to solve problems with decimals?

A: Yes, you can use geometric progressions to solve problems with decimals by using the formula for the nth term of a geometric progression and substituting the decimal for the common ratio.

Q: How do I use geometric progressions to solve problems with complex numbers?

A: You can use geometric progressions to solve problems with complex numbers by using the formula for the nth term of a geometric progression and substituting the complex number for the common ratio.

Conclusion

In this article, we have answered some common questions about geometric progressions, including how to find the common ratio, how to calculate the nth term, and how to use geometric progressions to solve real-world problems. We have also discussed some common applications of geometric progressions and how to determine if a sequence is a geometric progression.