Find The First Four Terms Of The Sequence Given By The Following:$\[ A_n = 2 \cdot (-3)^{n-1}, \quad N = 1, 2, 3, \ldots \\]

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 general formula for a geometric sequence is given by $a_n = a_1 \cdot r^{n-1}$, where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. In this article, we will explore the first four terms of a geometric sequence given by the formula $a_n = 2 \cdot (-3)^{n-1}$, where $n = 1, 2, 3, \ldots$.

Understanding the Formula

The given formula $a_n = 2 \cdot (-3)^{n-1}$ indicates that the first term $a_1$ is 2, and the common ratio $r$ is $-3$. The exponent $n-1$ means that we need to raise the common ratio to the power of $n-1$ to get the nth term. This formula is a geometric sequence because each term is obtained by multiplying the previous term by the common ratio.

Finding the First Four Terms

To find the first four terms of the sequence, we need to substitute $n = 1, 2, 3, 4$ into the formula $a_n = 2 \cdot (-3)^{n-1}$.

First Term ($n = 1$)

For $n = 1$, we have: $a_1 = 2 \cdot (-3)^{1-1}$ $a_1 = 2 \cdot (-3)^0$ $a_1 = 2 \cdot 1$ $a_1 = 2$

Second Term ($n = 2$)

For $n = 2$, we have: $a_2 = 2 \cdot (-3)^{2-1}$ $a_2 = 2 \cdot (-3)^1$ $a_2 = 2 \cdot (-3)$ $a_2 = -6$

Third Term ($n = 3$)

For $n = 3$, we have: $a_3 = 2 \cdot (-3)^{3-1}$ $a_3 = 2 \cdot (-3)^2$ $a_3 = 2 \cdot 9$ $a_3 = 18$

Fourth Term ($n = 4$)

For $n = 4$, we have: $a_4 = 2 \cdot (-3)^{4-1}$ $a_4 = 2 \cdot (-3)^3$ $a_4 = 2 \cdot (-27)$ $a_4 = -54$

Conclusion

In conclusion, the first four terms of the geometric sequence given by the formula $a_n = 2 \cdot (-3)^{n-1}$ are $2, -6, 18, -54$. These terms are obtained by substituting $n = 1, 2, 3, 4$ into the formula and simplifying the expressions.

Applications of Geometric Sequences

Geometric sequences have numerous applications in mathematics, science, and engineering. Some of the applications include:

• Finance: Geometric sequences are used to calculate compound interest and depreciation.
• Biology: Geometric sequences are used to model population growth and decline.
• Physics: Geometric sequences are used to describe the motion of objects under constant acceleration.
• Computer Science: Geometric sequences are used in algorithms for image processing and computer graphics.

Final Thoughts

In this article, we have explored the first four terms of a geometric sequence given by the formula $a_n = 2 \cdot (-3)^{n-1}$. We have also discussed the applications of geometric sequences in various fields. Geometric sequences are a fundamental concept in mathematics, and understanding them is essential for solving problems in science, engineering, and finance.

Introduction

In our previous article, we explored the first four terms of a geometric sequence given by the formula $a_n = 2 \cdot (-3)^{n-1}$. Geometric sequences are a fundamental concept in mathematics, and understanding them is essential for solving problems in science, engineering, and finance. In this article, we will answer some frequently asked questions about geometric sequences.

Q&A

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 given by $a_n = a_1 \cdot r^{n-1}$, where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

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

A: To find the first term of a geometric sequence, you need to know the value of the common ratio and the value of the nth term. You can use the formula $a_1 = \frac{a_n}{r^{n-1}}$ to find the first term.

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

A: To find the common ratio of a geometric sequence, you need to know the values of two consecutive terms. You can use the formula $r = \frac{a_{n+1}}{a_n}$ to find the common ratio.

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 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 find the sum of a geometric sequence?

A: To find the sum of a geometric sequence, you need to know the first term, the common ratio, and the number of terms. You can use the formula $S_n = \frac{a_1(1-r^n)}{1-r}$ to find the sum.

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

A: To find the nth term of a geometric sequence, you need to know the first term and the common ratio. You can use the formula $a_n = a_1 \cdot r^{n-1}$ to find the nth term.

Q: What is the application of geometric sequences in finance?

A: Geometric sequences are used to calculate compound interest and depreciation in finance. For example, if you invest $1000 at a 5% annual interest rate, the amount after one year will be $1050, after two years will be $1102.50, and so on.

Q: What is the application of geometric sequences in biology?

A: Geometric sequences are used to model population growth and decline in biology. For example, if a population of bacteria doubles every hour, the number of bacteria after one hour will be 2 times the initial population, after two hours will be 4 times the initial population, and so on.

Conclusion

In conclusion, geometric sequences are a fundamental concept in mathematics, and understanding them is essential for solving problems in science, engineering, and finance. We hope that this article has answered some of the frequently asked questions about geometric sequences and has provided a better understanding of this important mathematical concept.

Final Thoughts

Geometric sequences are a powerful tool for modeling real-world phenomena, and they have numerous applications in various fields. By understanding geometric sequences, you can solve problems in finance, biology, physics, and computer science. We hope that this article has inspired you to learn more about geometric sequences and to explore their many applications.