What Is The Value Of \[$ A_1 \$\] Of The Geometric Series?$\[ \sum_{n=1}^{\infty} 12\left(-\frac{1}{9}\right)^{n-1} \\]A. \[$-\frac{12}{9}\$\]B. \[$-\frac{1}{9}\$\]C. 1D. 12

by ADMIN 174 views

Introduction

A geometric series is a type of mathematical series that consists of the sum of the terms of a geometric sequence. The 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 value of { a_1 $}$ of the geometric series is the first term of the series. In this article, we will discuss how to find the value of { a_1 $}$ of the geometric series.

Understanding Geometric Series

A geometric series is defined as:

∑n=1∞a⋅rn−1\sum_{n=1}^{\infty} a \cdot r^{n-1}

where:

  • aa is the first term of the series
  • rr is the common ratio
  • nn is the term number

The value of { a_1 $}$ of the geometric series is the first term of the series, which is denoted by aa. The common ratio rr is the ratio of each term to its previous term.

Finding the Value of { a_1 $}$

To find the value of { a_1 $}$ of the geometric series, we need to identify the first term of the series. In the given series:

∑n=1∞12(−19)n−1\sum_{n=1}^{\infty} 12\left(-\frac{1}{9}\right)^{n-1}

The first term of the series is 12(−19)012\left(-\frac{1}{9}\right)^{0}, which is equal to 1212.

Calculating the Value of { a_1 $}$

The value of { a_1 $}$ of the geometric series is the first term of the series, which is 1212. Therefore, the value of { a_1 $}$ of the geometric series is:

a1=12a_1 = 12

Conclusion

In conclusion, the value of { a_1 $}$ of the geometric series is the first term of the series. To find the value of { a_1 $}$, we need to identify the first term of the series. In the given series, the first term is 1212. Therefore, the value of { a_1 $}$ of the geometric series is 1212.

Example Problems

Here are some example problems to help you understand how to find the value of { a_1 $}$ of the geometric series:

  • Find the value of { a_1 $}$ of the geometric series: ∑n=1∞5(12)n−1\sum_{n=1}^{\infty} 5\left(\frac{1}{2}\right)^{n-1}
  • Find the value of { a_1 $}$ of the geometric series: ∑n=1∞3(−14)n−1\sum_{n=1}^{\infty} 3\left(-\frac{1}{4}\right)^{n-1}
  • Find the value of { a_1 $}$ of the geometric series: ∑n=1∞2(13)n−1\sum_{n=1}^{\infty} 2\left(\frac{1}{3}\right)^{n-1}

Solutions

Here are the solutions to the example problems:

  • Find the value of { a_1 $}$ of the geometric series: ∑n=1∞5(12)n−1\sum_{n=1}^{\infty} 5\left(\frac{1}{2}\right)^{n-1} The first term of the series is 5(12)05\left(\frac{1}{2}\right)^{0}, which is equal to 55. Therefore, the value of { a_1 $}$ of the geometric series is 55.
  • Find the value of { a_1 $}$ of the geometric series: ∑n=1∞3(−14)n−1\sum_{n=1}^{\infty} 3\left(-\frac{1}{4}\right)^{n-1} The first term of the series is 3(−14)03\left(-\frac{1}{4}\right)^{0}, which is equal to 33. Therefore, the value of { a_1 $}$ of the geometric series is 33.
  • Find the value of { a_1 $}$ of the geometric series: ∑n=1∞2(13)n−1\sum_{n=1}^{\infty} 2\left(\frac{1}{3}\right)^{n-1} The first term of the series is 2(13)02\left(\frac{1}{3}\right)^{0}, which is equal to 22. Therefore, the value of { a_1 $}$ of the geometric series is 22.

Final Answer

Q: What is a geometric series?

A: A geometric series is a type of mathematical series that consists of the sum of the terms of a geometric sequence. The 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 value of { a_1 $}$ of a geometric series?

A: The value of { a_1 $}$ of a geometric series is the first term of the series. It is denoted by aa in the formula for the geometric series.

Q: How do I find the value of { a_1 $}$ of a geometric series?

A: To find the value of { a_1 $}$ of a geometric series, you need to identify the first term of the series. The first term is the term that is not multiplied by the common ratio.

Q: What is the formula for the geometric series?

A: The formula for the geometric series is:

∑n=1∞a⋅rn−1\sum_{n=1}^{\infty} a \cdot r^{n-1}

where:

  • aa is the first term of the series
  • rr is the common ratio
  • nn is the term number

Q: How do I calculate the value of { a_1 $}$ of a geometric series?

A: To calculate the value of { a_1 $}$ of a geometric series, you need to identify the first term of the series and substitute it into the formula for the geometric series.

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

A: The common ratio in a geometric series is the ratio of each term to its previous term. It is denoted by rr in the formula for the geometric series.

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

A: To find the common ratio of a geometric series, you need to divide each term by its previous term.

Q: What is the sum of an infinite geometric series?

A: The sum of an infinite geometric series is given by the formula:

S=a1−rS = \frac{a}{1 - r}

where:

  • aa is the first term of the series
  • rr is the common ratio

Q: How do I use the formula for the sum of an infinite geometric series?

A: To use the formula for the sum of an infinite geometric series, you need to substitute the values of aa and rr into the formula.

Q: What are some examples of geometric series?

A: Some examples of geometric series include:

  • ∑n=1∞5(12)n−1\sum_{n=1}^{\infty} 5\left(\frac{1}{2}\right)^{n-1}
  • ∑n=1∞3(−14)n−1\sum_{n=1}^{\infty} 3\left(-\frac{1}{4}\right)^{n-1}
  • ∑n=1∞2(13)n−1\sum_{n=1}^{\infty} 2\left(\frac{1}{3}\right)^{n-1}

Q: How do I find the value of { a_1 $}$ of each of these geometric series?

A: To find the value of { a_1 $}$ of each of these geometric series, you need to identify the first term of each series and substitute it into the formula for the geometric series.

Final Answer

The final answer is: 12\boxed{12}