This Is The Recursive Formula For A Geometric Sequence:$\[ \begin{align*} f(1) &= 8,000 \\ f(n) &= \frac{1}{2} F(n-1), \text{ For } N \geq 2 \end{align*} \\]What Is The Fifth Term In The Sequence?Select The Correct Answer:A. 250 B. 500 C.

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Introduction

In mathematics, a geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The recursive formula for a geometric sequence is given by:

{ \begin{align*} f(1) &= 8,000 \\ f(n) &= \frac{1}{2} f(n-1), \text{ for } n \geq 2 \end{align*} \}

In this article, we will use the recursive formula to find the fifth term in the sequence.

Understanding the Recursive Formula

The recursive formula for a geometric sequence is defined as:

f(n)=12f(n−1){ f(n) = \frac{1}{2} f(n-1) }

This formula states that each term in the sequence is half of the previous term. The first term in the sequence is given as f(1)=8,000f(1) = 8,000.

Finding the Second Term

To find the second term in the sequence, we can use the recursive formula:

f(2)=12f(1){ f(2) = \frac{1}{2} f(1) }

Substituting the value of f(1)=8,000f(1) = 8,000, we get:

f(2)=12×8,000=4,000{ f(2) = \frac{1}{2} \times 8,000 = 4,000 }

Finding the Third Term

To find the third term in the sequence, we can use the recursive formula:

f(3)=12f(2){ f(3) = \frac{1}{2} f(2) }

Substituting the value of f(2)=4,000f(2) = 4,000, we get:

f(3)=12×4,000=2,000{ f(3) = \frac{1}{2} \times 4,000 = 2,000 }

Finding the Fourth Term

To find the fourth term in the sequence, we can use the recursive formula:

f(4)=12f(3){ f(4) = \frac{1}{2} f(3) }

Substituting the value of f(3)=2,000f(3) = 2,000, we get:

f(4)=12×2,000=1,000{ f(4) = \frac{1}{2} \times 2,000 = 1,000 }

Finding the Fifth Term

To find the fifth term in the sequence, we can use the recursive formula:

f(5)=12f(4){ f(5) = \frac{1}{2} f(4) }

Substituting the value of f(4)=1,000f(4) = 1,000, we get:

f(5)=12×1,000=500{ f(5) = \frac{1}{2} \times 1,000 = 500 }

Conclusion

In this article, we used the recursive formula for a geometric sequence to find the fifth term in the sequence. We started with the first term f(1)=8,000f(1) = 8,000 and used the recursive formula to find each subsequent term. The fifth term in the sequence is f(5)=500f(5) = 500.

Answer

The correct answer is B. 500.

References

Discussion

Introduction

In our previous article, we explored the recursive formula for a geometric sequence and used it to find the fifth term in the sequence. In this article, we will answer some frequently asked questions about geometric sequences and the recursive formula.

Q: What is a geometric sequence?

A: A geometric sequence is a type of sequence 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 recursive formula for a geometric sequence?

A: The recursive formula for a geometric sequence is given by:

{ \begin{align*} f(1) &= 8,000 \\ f(n) &= \frac{1}{2} f(n-1), \text{ for } n \geq 2 \end{align*} \}

Q: How do I use the recursive formula to find the nth term in a geometric sequence?

A: To find the nth term in a geometric sequence, you can use the recursive formula:

f(n)=1rf(n−1){ f(n) = \frac{1}{r} f(n-1) }

where r is the common ratio.

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

A: The common ratio in a geometric sequence is the fixed, non-zero number that is multiplied by each term to get the next term.

Q: Can I use the recursive formula to find the sum of a geometric sequence?

A: Yes, you can use the recursive formula to find the sum of a geometric sequence. The sum of a geometric sequence is given by:

Sn=a(1−rn)1−r{ S_n = \frac{a(1-r^n)}{1-r} }

where a is the first term, r is the common ratio, and n is the number of terms.

Q: What is the difference between a geometric sequence and an arithmetic sequence?

A: A geometric sequence is a type of sequence 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, on the other hand, is a type of sequence where each term after the first is found by adding a fixed number called the common difference.

Q: Can I use the recursive formula to find the nth term in an arithmetic sequence?

A: No, you cannot use the recursive formula to find the nth term in an arithmetic sequence. The recursive formula is only applicable to geometric sequences.

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

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

  • Compound interest: Geometric sequences can be used to model the growth of an investment over time.
  • Population growth: Geometric sequences can be used to model the growth of a population over time.
  • Music: Geometric sequences can be used to model the frequency of notes in music.

Conclusion

In this article, we answered some frequently asked questions about geometric sequences and the recursive formula. We hope that this article has been helpful in understanding the concept of geometric sequences and the recursive formula.

References

Discussion

Do you have any questions about geometric sequences or the recursive formula? Share your thoughts and questions in the comments below!