Explain How To Find The A Th A^{\text{th}} A Th And N Th N^{\text{th}} N Th Terms Of A Geometric Progression (G.P.) Whose First Term Is 3 And Whose Common Ratio Is 1 3 \frac{1}{\sqrt{3}} 3 1 .

Mar 10, 2025 by ADMIN 197 views

$**Finding the $a^{\text{th}}$ and $n^{\text{th}}$ Terms of a Geometric Progression**$

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. In this article, we will discuss how to find the $a^{\text{th}}$ and $n^{\text{th}}$ terms of a geometric progression with a given first term and common ratio.

Understanding Geometric Progression

A geometric progression is a sequence of numbers that follows a specific pattern. The general form of a geometric progression is:

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

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

Formula for the $a^{\text{th}}$ Term

The formula for the $a^{\text{th}}$ term of a geometric progression is:

a_a = a \cdot r^{a-1}

where $a_a$ is the $a^{\text{th}}$ term, $a$ is the first term, and $r$ is the common ratio.

Formula for the $n^{\text{th}}$ Term

The formula for the $n^{\text{th}}$ term of a geometric progression is:

a_n = a \cdot r^{n-1}

where $a_n$ is the $n^{\text{th}}$ term, $a$ is the first term, and $r$ is the common ratio.

Example: Finding the $a^{\text{th}}$ and $n^{\text{th}}$ Terms

Let's consider a geometric progression with a first term of 3 and a common ratio of $\frac{1}{\sqrt{3}}$ . We want to find the $a^{\text{th}}$ and $n^{\text{th}}$ terms of this progression.

Using the formula for the $a^{\text{th}}$ term, we get:

a_a = 3 \cdot \left(\frac{1}{\sqrt{3}}\right)^{a-1}

Using the formula for the $n^{\text{th}}$ term, we get:

a_n = 3 \cdot \left(\frac{1}{\sqrt{3}}\right)^{n-1}

Calculating the $a^{\text{th}}$ Term

To calculate the $a^{\text{th}}$ term, we need to substitute the value of $a$ into the formula. Let's say we want to find the 5th term. Substituting $a=5$ into the formula, we get:

a_5 = 3 \cdot \left(\frac{1}{\sqrt{3}}\right)^{5-1}

Simplifying the expression, we get:

a_5 = 3 \cdot \left(\frac{1}{\sqrt{3}}\right)^4

a_5 = 3 \cdot \frac{1}{3\sqrt{3}}

a_5 = \frac{1}{\sqrt{3}}

Calculating the $n^{\text{th}}$ Term

To calculate the $n^{\text{th}}$ term, we need to substitute the value of $n$ into the formula. Let's say we want to find the 10th term. Substituting $n=10$ into the formula, we get:

a_{10} = 3 \cdot \left(\frac{1}{\sqrt{3}}\right)^{10-1}

Simplifying the expression, we get:

a_{10} = 3 \cdot \left(\frac{1}{\sqrt{3}}\right)^9

a_{10} = 3 \cdot \frac{1}{3^4\sqrt{3}}

a_{10} = \frac{1}{3^3\sqrt{3}}

Conclusion

In this article, we discussed how to find the $a^{\text{th}}$ and $n^{\text{th}}$ terms of a geometric progression with a given first term and common ratio. We used the formulas for the $a^{\text{th}}$ and $n^{\text{th}}$ terms to calculate the 5th and 10th terms of a geometric progression with a first term of 3 and a common ratio of $\frac{1}{\sqrt{3}}$ . We hope that this article has provided a clear understanding of how to find the $a^{\text{th}}$ and $n^{\text{th}}$ terms of a geometric progression.

References

[1] "Geometric Progression" by Math Open Reference
[2] "Geometric Progression Formula" by BYJU'S
[3] "Geometric Progression Examples" by Khan Academy

Frequently Asked 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 I find the $a^{\text{th}}$ term of a geometric progression? A: To find the $a^{\text{th}}$ term, use the formula: $a_a = a \cdot r^{a-1}$ .
Q: How do I find the $n^{\text{th}}$ term of a geometric progression? A: To find the $n^{\text{th}}$ term, use the formula: $a_n = a \cdot r^{n-1}$ .

Glossary

Geometric Progression: 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.
Common Ratio: The fixed, non-zero number that is used to multiply each term in a geometric progression.
First Term: The first number in a geometric progression.
$a^{\text{th}}$ Term: The term in a geometric progression that is $a$ positions from the first term.
$n^{\text{th}}$ Term: The term in a geometric progression that is $n$ positions from the first term.
Geometric Progression Q&A ==========================

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 I find the $a^{\text{th}}$ term of a geometric progression?

A: To find the $a^{\text{th}}$ term, use the formula: $a_a = a \cdot r^{a-1}$ , where $a$ is the first term and $r$ is the common ratio.

Q: How do I find the $n^{\text{th}}$ term of a geometric progression?

A: To find the $n^{\text{th}}$ term, use the formula: $a_n = a \cdot r^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.

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

A: The common ratio is the fixed, non-zero number that is used to multiply each term in a geometric progression.

Q: How do I calculate the common ratio of a geometric progression?

A: To calculate the common ratio, divide any term by the previous term. For example, if the first term is 3 and the second term is 6, the common ratio is 6/3 = 2.

Q: What is the first term of a geometric progression?

A: The first term is the first number in a geometric progression.

Q: How do I find the first term of a geometric progression?

A: To find the first term, look at the first number in the sequence.

Q: What is the difference between a geometric progression and an arithmetic 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. An arithmetic progression is a sequence of numbers where each term after the first is found by adding a fixed number called the common difference.

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

A: To determine if a sequence is a geometric progression or an arithmetic progression, look at the ratio between consecutive terms. If the ratio is constant, it is a geometric progression. If the difference between consecutive terms is constant, it is an arithmetic progression.

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

A: Some real-world examples of geometric progressions include:

Population growth: The population of a city may grow at a rate that is proportional to the current population.
Compound interest: The interest on a savings account may be compounded at a rate that is proportional to the current balance.
Sound waves: The amplitude of a sound wave may decrease at a rate that is proportional to the distance from the source.

Q: How do I use geometric progressions in real-world applications?

A: Geometric progressions can be used to model a wide range of real-world phenomena, including population growth, compound interest, and sound waves. To use geometric progressions in real-world applications, identify the first term and the common ratio, and then use the formula for the $a^{\text{th}}$ term or the $n^{\text{th}}$ term to calculate the desired value.

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

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

Assuming that the common ratio is always positive.
Failing to check for convergence or divergence.
Using the wrong formula for the $a^{\text{th}}$ term or the $n^{\text{th}}$ term.

Q: How do I check for convergence or divergence of a geometric progression?

A: To check for convergence or divergence, calculate the limit of the $a^{\text{th}}$ term or the $n^{\text{th}}$ term as $a$ or $n$ approaches infinity. If the limit is finite, the progression converges. If the limit is infinite, the progression diverges.

Q: What are some advanced topics in geometric progressions?

A: Some advanced topics in geometric progressions include:

Infinite geometric progressions.
Convergence and divergence of geometric progressions.
Geometric progressions with complex numbers.
Geometric progressions with matrices.

Q: How do I learn more about geometric progressions?

A: To learn more about geometric progressions, read books or articles on the subject, take online courses or watch video lectures, or practice solving problems and exercises.