Which Formula Can Be Used To Find The $n$th Term In A Geometric Sequence Where $a_1=3$ And $r=2$?A. $a_n=3+2(n-1)$ B. $a_n=3(n-1)+2$ C. $a_n=3^{n-1} \cdot 2$ D. $a_n=3 \cdot

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Introduction

In mathematics, a geometric sequence 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. The formula for finding the nth term in a geometric sequence is essential in various mathematical and real-world applications. In this article, we will explore the formula for finding the nth term in a geometric sequence and apply it to a specific problem.

Understanding Geometric Sequences

A geometric sequence 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. The general formula for a geometric sequence is:

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}

where:

  • a_n$ is the nth term in the sequence

  • a_1$ is the first term in the sequence

  • r$ is the common ratio

  • n$ is the term number

The Problem

We are given a geometric sequence with $a_1=3$ and $r=2$. We need to find the formula for the nth term in this sequence.

Analyzing the Options

Let's analyze the options given:

A. $a_n=3+2(n-1)$ B. $a_n=3(n-1)+2$ C. $a_n=3^{n-1} \cdot 2$ D. $a_n=3 \cdot 2^{n-1}$

Option A: $a_n=3+2(n-1)$

This option is incorrect because it does not take into account the common ratio $r$. The formula for a geometric sequence must include the common ratio raised to the power of $n-1$.

Option B: $a_n=3(n-1)+2$

This option is also incorrect because it does not take into account the common ratio $r$. The formula for a geometric sequence must include the common ratio raised to the power of $n-1$.

Option C: $a_n=3^{n-1} \cdot 2$

This option is incorrect because it uses the first term $a_1$ raised to the power of $n-1$, instead of the common ratio $r$ raised to the power of $n-1$.

Option D: $a_n=3 \cdot 2^{n-1}$

This option is correct because it uses the first term $a_1$ multiplied by the common ratio $r$ raised to the power of $n-1$.

Conclusion

In conclusion, the correct formula for finding the nth term in a geometric sequence where $a_1=3$ and $r=2$ is:

an=3⋅2n−1a_n=3 \cdot 2^{n-1}

This formula takes into account the first term $a_1$ and the common ratio $r$, and is essential in various mathematical and real-world applications.

Real-World Applications

Geometric sequences have numerous real-world applications, including:

  • Finance: Geometric sequences are used to calculate compound interest and investment returns.
  • Biology: Geometric sequences are used to model population growth and decline.
  • Computer Science: Geometric sequences are used in algorithms for image processing and computer graphics.

Final Thoughts

In this article, we explored the formula for finding the nth term in a geometric sequence and applied it to a specific problem. We analyzed the options given and concluded that the correct formula is:

an=3⋅2n−1a_n=3 \cdot 2^{n-1}

Introduction

In our previous article, we explored the formula for finding the nth term in a geometric sequence and applied it to a specific problem. In this article, we will answer some frequently asked questions about geometric sequences and provide additional insights into this important mathematical concept.

Q: What is a geometric sequence?

A: A geometric sequence 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 formula for a geometric sequence?

A: The formula for a geometric sequence is:

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}

where:

  • a_n$ is the nth term in the sequence

  • a_1$ is the first term in the sequence

  • r$ is the common ratio

  • n$ is the term number

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

A: To find the common ratio in a geometric sequence, you can divide any term by its previous term. For example, if the sequence is 2, 6, 18, 54, ..., you can divide the second term by the first term to get the common ratio:

62=3\frac{6}{2} = 3

Q: How do I find the nth term in a geometric sequence?

A: To find the nth term in a geometric sequence, you can use the formula:

an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}

where:

  • a_n$ is the nth term in the sequence

  • a_1$ is the first term in the sequence

  • r$ is the common ratio

  • n$ is the term number

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

A: A geometric sequence 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. An arithmetic sequence, on the other hand, 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 geometric or arithmetic?

A: To determine if a sequence is geometric or arithmetic, you can look at the relationship between the terms. If the terms are obtained by multiplying the previous term by a fixed number, the sequence is geometric. If the terms are obtained by adding a fixed number to the previous term, the sequence is arithmetic.

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

A: Geometric sequences have numerous real-world applications, including:

  • Finance: Geometric sequences are used to calculate compound interest and investment returns.
  • Biology: Geometric sequences are used to model population growth and decline.
  • Computer Science: Geometric sequences are used in algorithms for image processing and computer graphics.

Q: How do I use geometric sequences in finance?

A: Geometric sequences are used in finance to calculate compound interest and investment returns. For example, if you invest $100 at a 5% annual interest rate, the amount after one year will be:

100â‹…(1+0.05)=105100 \cdot (1 + 0.05) = 105

After two years, the amount will be:

105â‹…(1+0.05)=110.25105 \cdot (1 + 0.05) = 110.25

And so on.

Q: How do I use geometric sequences in biology?

A: Geometric sequences are used in biology to model population growth and decline. For example, if a population of bacteria grows at a rate of 20% per day, the population after one day will be:

100â‹…(1+0.20)=120100 \cdot (1 + 0.20) = 120

After two days, the population will be:

120â‹…(1+0.20)=144120 \cdot (1 + 0.20) = 144

And so on.

Conclusion

In this article, we answered some frequently asked questions about geometric sequences and provided additional insights into this important mathematical concept. We hope this article has been helpful in understanding geometric sequences and their applications.