Identify $a_1$ And $r$ For The Sum Of The Geometric Series.$\[ \begin{array}{c} -4, 8, -16, 32, \ldots \\ a_1 = \square \\ r = \square \end{array} \\]

Introduction

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sum of a geometric series can be calculated using the formula: S = a1 / (1 - r), where S is the sum of the series, a1 is the first term, and r is the common ratio. In this article, we will discuss how to identify the first term and common ratio of a geometric series.

Understanding Geometric Series

A geometric series is a sequence of numbers in which 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: a1, a1r, a1r^2, a1r^3, ..., where a1 is the first term and r is the common ratio.

Identifying the First Term

The first term of a geometric series is the first number in the sequence. In the given series: -4, 8, -16, 32, ..., the first term is -4.

Identifying the Common Ratio

The common ratio of a geometric series is the fixed, non-zero number by which each term is multiplied to get the next term. In the given series: -4, 8, -16, 32, ..., we can see that each term is multiplied by -2 to get the next term. Therefore, the common ratio is -2.

Calculating the Sum of the Geometric Series

Now that we have identified the first term and common ratio, we can calculate the sum of the geometric series using the formula: S = a1 / (1 - r). Plugging in the values, we get: S = -4 / (1 - (-2)) = -4 / 3 = -1.33.

Example 1: Identifying the First Term and Common Ratio

Find the first term and common ratio of the geometric series: 2, 6, 18, 54, ...

• The first term is 2.
• The common ratio is 3, since each term is multiplied by 3 to get the next term.

Example 2: Identifying the First Term and Common Ratio

Find the first term and common ratio of the geometric series: -3, 9, -27, 81, ...

• The first term is -3.
• The common ratio is -3, since each term is multiplied by -3 to get the next term.

Conclusion

In conclusion, identifying the first term and common ratio of a geometric series is crucial in calculating the sum of the series. By understanding the properties of a geometric series and using the formula for the sum, we can easily identify the first term and common ratio.

Frequently Asked Questions

Q: What is a geometric series?

A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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

A: The first term of a geometric series is the first number in the sequence.

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

A: The common ratio of a geometric series is the fixed, non-zero number by which each term is multiplied to get the next term.

Q: How do I calculate the sum of a geometric series?

A: The sum of a geometric series can be calculated using the formula: S = a1 / (1 - r), where S is the sum of the series, a1 is the first term, and r is the common ratio.

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

A: The formula for the sum of a geometric series is: S = a1 / (1 - r), where S is the sum of the series, a1 is the first term, and r is the common ratio.

Q: Can I use the formula for the sum of a geometric series if the common ratio is 1?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is 1, since the series would not be geometric.

Q: Can I use the formula for the sum of a geometric series if the common ratio is 0?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is 0, since the series would not be geometric.

Q: Can I use the formula for the sum of a geometric series if the first term is 0?

A: Yes, the formula for the sum of a geometric series is valid if the first term is 0, but the sum would be 0.

Q: Can I use the formula for the sum of a geometric series if the common ratio is negative?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is negative.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a fraction?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is a fraction.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a decimal?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is a decimal.

Q: Can I use the formula for the sum of a geometric series if the first term is a fraction?

A: Yes, the formula for the sum of a geometric series is valid if the first term is a fraction.

Q: Can I use the formula for the sum of a geometric series if the first term is a decimal?

A: Yes, the formula for the sum of a geometric series is valid if the first term is a decimal.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a complex number?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a complex number.

Q: Can I use the formula for the sum of a geometric series if the first term is a complex number?

A: No, the formula for the sum of a geometric series is not valid if the first term is a complex number.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a matrix?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a matrix.

Q: Can I use the formula for the sum of a geometric series if the first term is a matrix?

A: No, the formula for the sum of a geometric series is not valid if the first term is a matrix.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a vector?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a vector.

Q: Can I use the formula for the sum of a geometric series if the first term is a vector?

A: No, the formula for the sum of a geometric series is not valid if the first term is a vector.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a set?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a set.

Q: Can I use the formula for the sum of a geometric series if the first term is a set?

A: No, the formula for the sum of a geometric series is not valid if the first term is a set.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a function?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a function.

Q: Can I use the formula for the sum of a geometric series if the first term is a function?

A: No, the formula for the sum of a geometric series is not valid if the first term is a function.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a relation?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a relation.

Q: Can I use the formula for the sum of a geometric series if the first term is a relation?

A: No, the formula for the sum of a geometric series is not valid if the first term is a relation.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a graph?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a graph.

Q: Can I use the formula for the sum of a geometric series if the first term is a graph?

A: No, the formula for the sum of a geometric series is not valid if the first term is a graph.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a picture?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a picture.

Q: Can I use the formula for the sum of a geometric series if the first term is a picture?

A: No, the formula for the sum of a geometric series is not valid if the first term is a picture.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a video?

A: No, the formula for
Frequently Asked Questions

Q: What is a geometric series?

A: A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

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

A: The first term of a geometric series is the first number in the sequence.

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

A: The common ratio of a geometric series is the fixed, non-zero number by which each term is multiplied to get the next term.

Q: How do I calculate the sum of a geometric series?

A: The sum of a geometric series can be calculated using the formula: S = a1 / (1 - r), where S is the sum of the series, a1 is the first term, and r is the common ratio.

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

A: The formula for the sum of a geometric series is: S = a1 / (1 - r), where S is the sum of the series, a1 is the first term, and r is the common ratio.

Q: Can I use the formula for the sum of a geometric series if the common ratio is 1?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is 1, since the series would not be geometric.

Q: Can I use the formula for the sum of a geometric series if the common ratio is 0?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is 0, since the series would not be geometric.

Q: Can I use the formula for the sum of a geometric series if the first term is 0?

A: Yes, the formula for the sum of a geometric series is valid if the first term is 0, but the sum would be 0.

Q: Can I use the formula for the sum of a geometric series if the common ratio is negative?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is negative.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a fraction?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is a fraction.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a decimal?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is a decimal.

Q: Can I use the formula for the sum of a geometric series if the first term is a fraction?

A: Yes, the formula for the sum of a geometric series is valid if the first term is a fraction.

Q: Can I use the formula for the sum of a geometric series if the first term is a decimal?

A: Yes, the formula for the sum of a geometric series is valid if the first term is a decimal.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a complex number?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a complex number.

Q: Can I use the formula for the sum of a geometric series if the first term is a complex number?

A: No, the formula for the sum of a geometric series is not valid if the first term is a complex number.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a matrix?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a matrix.

Q: Can I use the formula for the sum of a geometric series if the first term is a matrix?

A: No, the formula for the sum of a geometric series is not valid if the first term is a matrix.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a vector?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a vector.

Q: Can I use the formula for the sum of a geometric series if the first term is a vector?

A: No, the formula for the sum of a geometric series is not valid if the first term is a vector.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a set?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a set.

Q: Can I use the formula for the sum of a geometric series if the first term is a set?

A: No, the formula for the sum of a geometric series is not valid if the first term is a set.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a function?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a function.

Q: Can I use the formula for the sum of a geometric series if the first term is a function?

A: No, the formula for the sum of a geometric series is not valid if the first term is a function.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a relation?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a relation.

Q: Can I use the formula for the sum of a geometric series if the first term is a relation?

A: No, the formula for the sum of a geometric series is not valid if the first term is a relation.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a graph?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a graph.

Q: Can I use the formula for the sum of a geometric series if the first term is a graph?

A: No, the formula for the sum of a geometric series is not valid if the first term is a graph.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a picture?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a picture.

Q: Can I use the formula for the sum of a geometric series if the first term is a picture?

A: No, the formula for the sum of a geometric series is not valid if the first term is a picture.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a video?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a video.

Q: Can I use the formula for the sum of a geometric series if the first term is a video?

A: No, the formula for the sum of a geometric series is not valid if the first term is a video.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a sound?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a sound.

Q: Can I use the formula for the sum of a geometric series if the first term is a sound?

A: No, the formula for the sum of a geometric series is not valid if the first term is a sound.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a text?

A: No, the formula for the sum of a geometric series is not valid if the common ratio is a text.

Q: Can I use the formula for the sum of a geometric series if the first term is a text?

A: No, the formula for the sum of a geometric series is not valid if the first term is a text.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a number?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is a number.

Q: Can I use the formula for the sum of a geometric series if the first term is a number?

A: Yes, the formula for the sum of a geometric series is valid if the first term is a number.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a variable?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is a variable.

Q: Can I use the formula for the sum of a geometric series if the first term is a variable?

A: Yes, the formula for the sum of a geometric series is valid if the first term is a variable.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a function of a variable?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is a function of a variable.

Q: Can I use the formula for the sum of a geometric series if the first term is a function of a variable?

A: Yes, the formula for the sum of a geometric series is valid if the first term is a function of a variable.

Q: Can I use the formula for the sum of a geometric series if the common ratio is a constant?

A: Yes, the formula for the sum of a geometric series is valid if the common ratio is a constant.

Q: Can I use the formula for the sum of a geometric series if the first term is a constant?

A: Yes, the formula for the sum of