Recursive Formulas in Geometric Sequences

Oct 13, 2025 by leolavanderia.com.br 76 views

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Introduction

Recursive formulas are a powerful tool in mathematics, particularly in the study of sequences and series. A recursive formula is a formula that defines a sequence recursively, meaning that each term is defined in terms of previous terms. In this article, we will explore the concept of recursive formulas in geometric sequences, with a focus on the given problem involving Sebastian's formula.

What is a Geometric Sequence?

A geometric sequence 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 sequence is:

a, ar, ar^2, ar^3, ...

where a is the first term and r is the common ratio.

Recursive Formulas in Geometric Sequences

A recursive formula for a geometric sequence can be written in the form:

f(x+1) = rf(x)

where f(x) is the nth term of the sequence, and r is the common ratio.

Sebastian's Formula

Sebastian writes the recursive formula:

f(x+1) = 4f(x)

to represent a geometric sequence whose second term is 12. We need to find the correct formula for f(x).

Finding the Correct Formula

To find the correct formula, we need to use the given information that the second term is 12. We can write the second term as:

f(2) = 12

Using the recursive formula, we can write:

f(2) = 4f(1)

Since f(2) = 12, we can substitute this value into the equation:

12 = 4f(1)

Dividing both sides by 4, we get:

f(1) = 3

Now that we have found the first term, we can use the recursive formula to find the general formula for f(x). We can write:

f(x+1) = 4f(x)

Substituting x = 1, we get:

f(2) = 4f(1)

Substituting f(1) = 3, we get:

f(2) = 4(3)

f(2) = 12

This confirms that the second term is indeed 12.

General Formula for f(x)

To find the general formula for f(x), we can use the recursive formula and the fact that f(1) = 3. We can write:

f(x+1) = 4f(x)

Substituting x = 1, we get:

f(2) = 4f(1)

Substituting f(1) = 3, we get:

f(2) = 4(3)

f(2) = 12

We can continue this process to find the general formula for f(x). We can write:

f(3) = 4f(2)

Substituting f(2) = 12, we get:

f(3) = 4(12)

f(3) = 48

We can continue this process to find the general formula for f(x). We can write:

f(x) = 4^x f(1)

Substituting f(1) = 3, we get:

f(x) = 4^x (3)

f(x) = 3(4^x)

This is the correct formula for f(x).

Conclusion

In this article, we explored the concept of recursive formulas in geometric sequences, with a focus on the given problem involving Sebastian's formula. We found that the correct formula for f(x) is:

f(x) = 3(4^x)

This formula represents a geometric sequence whose second term is 12.

Answer

The correct formula for f(x) is:

f(x) = 12(4)^x

This is option A.

Final Thoughts

Recursive formulas are a powerful tool in mathematics, particularly in the study of sequences and series. By understanding how to use recursive formulas, we can solve a wide range of problems involving geometric sequences. In this article, we saw how to use a recursive formula to find the general formula for f(x) in a geometric sequence. We also saw how to use the given information that the second term is 12 to find the correct formula for f(x).

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Introduction

In our previous article, we explored the concept of recursive formulas in geometric sequences, with a focus on the given problem involving Sebastian's formula. We found that the correct formula for f(x) is:

f(x) = 3(4^x)

In this article, we will continue to explore the concept of recursive formulas in geometric sequences, and answer some common questions that students may have.

Q&A

Q: What is a recursive formula?

A: A recursive formula is a formula that defines a sequence recursively, meaning that each term is defined in terms of previous terms.

Q: How do I know if a sequence is a geometric sequence?

A: A sequence is a geometric sequence if 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 in a geometric sequence?

A: To find the common ratio, you can divide any term by the previous term. For example, if the first term is 3 and the second term is 12, the common ratio is 12/3 = 4.

Q: How do I use a recursive formula to find the general formula for f(x)?

A: To use a recursive formula to find the general formula for f(x), you can start by writing the recursive formula:

f(x+1) = rf(x)

where f(x) is the nth term of the sequence, and r is the common ratio.

You can then substitute x = 1, 2, 3, ... to find the first few terms of the sequence. For example, if the recursive formula is:

f(x+1) = 4f(x)

you can substitute x = 1 to get:

f(2) = 4f(1)

You can then substitute x = 2 to get:

f(3) = 4f(2)

You can continue this process to find the general formula for f(x).

Q: How do I know if a recursive formula is correct?

A: To know if a recursive formula is correct, you can use the given information to check if the formula is true. For example, if the recursive formula is:

f(x+1) = 4f(x)

and the second term is 12, you can substitute x = 1 to get:

f(2) = 4f(1)

You can then substitute f(1) = 3 to get:

f(2) = 4(3)

f(2) = 12

This confirms that the recursive formula is correct.

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

A: Yes, you can use a recursive formula to find the sum of a geometric sequence. The sum of a geometric sequence can be found using the formula:

S_n = a(1 - r^n) / (1 - r)

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

You can use a recursive formula to find the general formula for f(x), and then use the formula for the sum of a geometric sequence to find the sum of the sequence.

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

A: Yes, you can use a recursive formula to find the product of a geometric sequence. The product of a geometric sequence can be found using the formula:

P_n = a^n

where P_n is the product of the first n terms, a is the first term, and n is the number of terms.

You can use a recursive formula to find the general formula for f(x), and then use the formula for the product of a geometric sequence to find the product of the sequence.

Conclusion

In this article, we answered some common questions that students may have about recursive formulas in geometric sequences. We also saw how to use a recursive formula to find the general formula for f(x), and how to use the formula for the sum and product of a geometric sequence to find the sum and product of the sequence.

Final Thoughts

Recursive formulas are a powerful tool in mathematics, particularly in the study of sequences and series. By understanding how to use recursive formulas, you can solve a wide range of problems involving geometric sequences. In this article, we saw how to use a recursive formula to find the general formula for f(x), and how to use the formula for the sum and product of a geometric sequence to find the sum and product of the sequence.