Given The Geometric Sequence: { -880, -220, -55, \ldots$}$What Is The Next Term In The Sequence?

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Introduction

Geometric sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, calculus, and other branches of 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. In this article, we will explore the concept of geometric sequences and use it to find the next term in a given sequence.

What is a Geometric Sequence?

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:

a_n = a_1 * r^(n-1)

where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.

Example of a Geometric Sequence

Let's consider an example of a geometric sequence: 2, 6, 18, 54, ...

In this sequence, the first term is 2, and the common ratio is 3. To find the next term, we multiply the previous term by the common ratio:

2 * 3 = 6 6 * 3 = 18 18 * 3 = 54 54 * 3 = 162

Therefore, the next term in the sequence is 162.

Finding the Next Term in the Given Sequence

Now, let's apply the concept of geometric sequences to the given sequence: -880, -220, -55, ...

To find the next term, we need to find the common ratio between the terms. We can do this by dividing each term by the previous term:

-220 / -880 = 1/4 -55 / -220 = 1/4

Since the common ratio is 1/4, we can multiply the last term by the common ratio to find the next term:

-55 * 1/4 = -13.75

Therefore, the next term in the sequence is -13.75.

Conclusion

In this article, we explored the concept of geometric sequences and used it to find the next term in a given sequence. Geometric sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, calculus, and other branches of mathematics. By applying the concept of geometric sequences, we can find the next term in a sequence and solve various problems in mathematics.

Common Ratio

The common ratio is a crucial concept in geometric sequences. It is the fixed, non-zero number that is multiplied by each term to get the next term. In the given sequence, the common ratio is 1/4.

Geometric Sequence Formula

The general formula for a geometric sequence is:

a_n = a_1 * r^(n-1)

where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.

Real-World Applications

Geometric sequences have numerous real-world applications. They are used in finance to calculate compound interest, in physics to describe the motion of objects, and in computer science to model population growth.

Tips and Tricks

  • To find the common ratio, divide each term by the previous term.
  • To find the next term, multiply the last term by the common ratio.
  • Geometric sequences can be used to model population growth, compound interest, and other real-world phenomena.

Frequently Asked Questions

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: How do I find the common ratio? A: To find the common ratio, divide each term by the previous term.

Q: How do I find the next term in a geometric sequence? A: To find the next term, multiply the last term by the common ratio.

References

  • "Geometric Sequences" by Math Open Reference
  • "Geometric Sequences" by Khan Academy
  • "Geometric Sequences" by Wolfram MathWorld
    Geometric Sequences Q&A: Frequently Asked Questions and Answers ================================================================

Introduction

Geometric sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will answer some of the most 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: How do I find the common ratio?

A: To find the common ratio, divide each term by the previous term. For example, if the sequence is 2, 6, 18, 54, ..., you can find the common ratio by dividing each term by the previous term:

2 / 6 = 1/3 6 / 18 = 1/3 18 / 54 = 1/3

Therefore, the common ratio is 1/3.

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

A: To find the next term, multiply the last term by the common ratio. For example, if the sequence is 2, 6, 18, 54, ..., the common ratio is 1/3, and the next term is:

54 * 1/3 = 18

Therefore, the next term in the sequence is 18.

Q: What is the formula for a geometric sequence?

A: The general formula for a geometric sequence is:

a_n = a_1 * r^(n-1)

where a_n is the nth term of the sequence, a_1 is the first term, r is the common ratio, and n is the term number.

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

A: To find the nth term, use the formula:

a_n = a_1 * r^(n-1)

For example, if the sequence is 2, 6, 18, 54, ..., the first term is 2, the common ratio is 1/3, and we want to find the 5th term. Plugging in the values, we get:

a_5 = 2 * (1/3)^(5-1) a_5 = 2 * (1/3)^4 a_5 = 2 * 1/81 a_5 = 2/81

Therefore, the 5th term in the sequence is 2/81.

Q: What is the sum of a geometric sequence?

A: The sum of a geometric sequence is given by the formula:

S_n = 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: How do I find the sum of a geometric sequence?

A: To find the sum, use the formula:

S_n = a_1 * (1 - r^n) / (1 - r)

For example, if the sequence is 2, 6, 18, 54, ..., the first term is 2, the common ratio is 1/3, and we want to find the sum of the first 5 terms. Plugging in the values, we get:

S_5 = 2 * (1 - (1/3)^5) / (1 - 1/3) S_5 = 2 * (1 - 1/243) / (2/3) S_5 = 2 * (242/243) / (2/3) S_5 = 242/243 * 3/2 S_5 = 121/81

Therefore, the sum of the first 5 terms in the sequence is 121/81.

Q: What is the product of a geometric sequence?

A: The product of a geometric sequence is given by the formula:

P_n = a_1 * r^(n-1)

where P_n is the product of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms.

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

A: To find the product, use the formula:

P_n = a_1 * r^(n-1)

For example, if the sequence is 2, 6, 18, 54, ..., the first term is 2, the common ratio is 1/3, and we want to find the product of the first 5 terms. Plugging in the values, we get:

P_5 = 2 * (1/3)^(5-1) P_5 = 2 * (1/3)^4 P_5 = 2 * 1/81 P_5 = 2/81

Therefore, the product of the first 5 terms in the sequence is 2/81.

Conclusion

In this article, we have answered some of the most frequently asked questions about geometric sequences. Geometric sequences are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, calculus, and other branches of mathematics. By applying the concepts and formulas discussed in this article, you can solve various problems involving geometric sequences.

References

  • "Geometric Sequences" by Math Open Reference
  • "Geometric Sequences" by Khan Academy
  • "Geometric Sequences" by Wolfram MathWorld