What Are The Values Of $a_1$ And $r$ Of The Geometric Series?$2, -2, 2, -2, 2$A. $ A 1 = 2 A_1=2 A 1 ​ = 2 [/tex] And $r=-2$B. $a_1=-2$ And $ R = 2 R=2 R = 2 [/tex]C. $a_1=-1$ And

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A geometric series 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 form of a geometric series is $a_1, a_1r, a_1r^2, a_1r^3, \ldots$, where $a_1$ is the first term and $r$ is the common ratio.

In this article, we will explore the values of $a_1$ and $r$ for the given geometric series: $2, -2, 2, -2, 2$.

What are Geometric Series?

A geometric series 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 form of a geometric series is $a_1, a_1r, a_1r^2, a_1r^3, \ldots$, where $a_1$ is the first term and $r$ is the common ratio.

Identifying the Values of $a_1$ and $r$

To identify the values of $a_1$ and $r$, we need to examine the given geometric series: $2, -2, 2, -2, 2$.

The first term of the series is $2$, which means that $a_1 = 2$.

To find the common ratio $r$, we can divide any term by its previous term. For example, we can divide the second term $-2$ by the first term $2$:

−22=−1\frac{-2}{2} = -1

However, this is not the common ratio. The common ratio is the number that we multiply the previous term by to get the next term. In this case, we can see that the second term $-2$ is obtained by multiplying the first term $2$ by $-1$, but the third term $2$ is obtained by multiplying the second term $-2$ by $-1$, and so on.

Therefore, the common ratio $r$ is $-1$.

Conclusion

In conclusion, the values of $a_1$ and $r$ for the given geometric series $2, -2, 2, -2, 2$ are:

a_1 = 2$ and $r = -1

This means that the correct answer is:

A. $a_1 = 2$ and $r = -1$

Example Use Case

Geometric series are used in many real-world applications, such as finance, economics, and engineering. For example, in finance, a geometric series can be used to model the growth of an investment over time. In economics, a geometric series can be used to model the growth of a population over time. In engineering, a geometric series can be used to model the behavior of a system over time.

Step-by-Step Solution

To solve this problem, we need to follow these steps:

  1. Identify the first term of the series.
  2. Divide any term by its previous term to find the common ratio.
  3. Check if the common ratio is correct by multiplying the previous term by the common ratio to get the next term.
  4. If the common ratio is correct, then the values of $a_1$ and $r$ are correct.

Common Mistakes

When solving geometric series problems, there are several common mistakes that students make. These include:

  • Not identifying the first term of the series.
  • Not checking if the common ratio is correct.
  • Not using the correct formula for the sum of a geometric series.

Tips and Tricks

When solving geometric series problems, here are some tips and tricks to keep in mind:

  • Always identify the first term of the series.
  • Always check if the common ratio is correct.
  • Always use the correct formula for the sum of a geometric series.

Conclusion

In conclusion, the values of $a_1$ and $r$ for the given geometric series $2, -2, 2, -2, 2$ are:

a_1 = 2$ and $r = -1

This means that the correct answer is:

In this article, we will answer some of the most frequently asked questions about geometric series.

Q: What is a geometric series?

A: A geometric series 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 are the two main components of a geometric series?

A: The two main components of a geometric series are the first term ($a_1$) and the common ratio ($r$).

Q: How do I identify the first term of a geometric series?

A: To identify the first term of a geometric series, simply look at the first number in the sequence.

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

A: To find the common ratio of a geometric series, divide any term by its previous term.

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

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

Sn=a1(1−rn)1−rS_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: What is the formula for the nth term of a geometric series?

A: The formula for the nth term of a geometric series is:

an=a1rn−1a_n = a_1r^{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 determine if a series is geometric?

A: To determine if a series is geometric, look for a pattern where each term is obtained by multiplying the previous term by a fixed, non-zero number.

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

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

  • Not identifying the first term of the series.
  • Not checking if the common ratio is correct.
  • Not using the correct formula for the sum of a geometric series.

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

A: Geometric series have many real-world applications, including:

  • Finance: Geometric series can be used to model the growth of an investment over time.
  • Economics: Geometric series can be used to model the growth of a population over time.
  • Engineering: Geometric series can be used to model the behavior of a system over time.

Q: How do I use geometric series in finance?

A: Geometric series can be used in finance to model the growth of an investment over time. For example, if you invest $100 at a 5% annual interest rate, the value of the investment after 1 year will be $105, after 2 years will be $110.25, and so on.

Q: How do I use geometric series in economics?

A: Geometric series can be used in economics to model the growth of a population over time. For example, if a population grows at a rate of 2% per year, the population after 1 year will be 102% of the original population, after 2 years will be 104.04% of the original population, and so on.

Q: How do I use geometric series in engineering?

A: Geometric series can be used in engineering to model the behavior of a system over time. For example, if a system decays at a rate of 10% per year, the value of the system after 1 year will be 90% of the original value, after 2 years will be 81% of the original value, and so on.

Conclusion

In conclusion, geometric series are a powerful tool for modeling real-world phenomena. By understanding the basics of geometric series, you can apply them to a wide range of fields, including finance, economics, and engineering.