The Formula $f(x+1)=\frac{2}{3}(f(x)$\] Defines A Geometric Sequence Where $f(1)=18$. Which Explicit Formula Can Be Used To Model The Same Sequence?A. $f(x)=(18)\left[\frac{2}{3}\right](x-1$\]B.

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Introduction

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 formula f(x+1)=23(f(x)f(x+1)=\frac{2}{3}(f(x) defines a geometric sequence, and we are given that f(1)=18f(1)=18. In this article, we will explore how to find the explicit formula for this sequence.

Understanding Geometric Sequences

A geometric sequence is defined by the formula an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}, where ana_n is the nth term of the sequence, a1a_1 is the first term, rr is the common ratio, and nn is the term number. In our case, the formula f(x+1)=23(f(x)f(x+1)=\frac{2}{3}(f(x) can be rewritten as f(x+1)=f(x)⋅23f(x+1) = f(x) \cdot \frac{2}{3}, which shows that the common ratio is 23\frac{2}{3}.

Finding the Explicit Formula

To find the explicit formula for this sequence, we can use the formula for a geometric sequence. We are given that f(1)=18f(1)=18, which means that a1=18a_1 = 18. We also know that the common ratio is 23\frac{2}{3}. Plugging these values into the formula for a geometric sequence, we get:

f(x)=18⋅(23)(x−1)f(x) = 18 \cdot \left(\frac{2}{3}\right)^{(x-1)}

This is the explicit formula for the sequence.

Verifying the Formula

To verify that this formula is correct, we can plug in some values for xx and see if we get the correct values for f(x)f(x). For example, if we plug in x=2x=2, we get:

f(2)=18⋅(23)(2−1)f(2) = 18 \cdot \left(\frac{2}{3}\right)^{(2-1)} f(2)=18⋅(23)1f(2) = 18 \cdot \left(\frac{2}{3}\right)^{1} f(2)=18⋅23f(2) = 18 \cdot \frac{2}{3} f(2)=12f(2) = 12

This shows that the formula is correct.

Conclusion

In this article, we have found the explicit formula for a geometric sequence defined by the formula f(x+1)=23(f(x)f(x+1)=\frac{2}{3}(f(x), where f(1)=18f(1)=18. The explicit formula is f(x)=18⋅(23)(x−1)f(x) = 18 \cdot \left(\frac{2}{3}\right)^{(x-1)}. We have also verified that this formula is correct by plugging in some values for xx.

Answer

The explicit formula for the sequence is:

f(x)=18⋅(23)(x−1)f(x) = 18 \cdot \left(\frac{2}{3}\right)^{(x-1)}

This is option A.

Discussion

The formula f(x+1)=23(f(x)f(x+1)=\frac{2}{3}(f(x) defines a geometric sequence, and we are given that f(1)=18f(1)=18. In this article, we have found the explicit formula for this sequence. The explicit formula is f(x)=18⋅(23)(x−1)f(x) = 18 \cdot \left(\frac{2}{3}\right)^{(x-1)}. We have also verified that this formula is correct by plugging in some values for xx.

References

  • [1] "Geometric Sequences". Math Open Reference. Retrieved 2023-02-26.
  • [2] "Geometric Sequences". Khan Academy. Retrieved 2023-02-26.

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Introduction

In our previous article, we explored the formula for a geometric sequence and found the explicit formula for a sequence defined by the formula f(x+1)=23(f(x)f(x+1)=\frac{2}{3}(f(x), where f(1)=18f(1)=18. In this article, we will answer some common questions about geometric sequences and explicit formulas.

Q&A

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 formula for a geometric sequence?

A: The formula for a geometric sequence is an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}, where ana_n is the nth term of the sequence, a1a_1 is the first term, rr is the common ratio, and nn is the term number.

Q: How do I find the explicit formula for a geometric sequence?

A: To find the explicit formula for a geometric sequence, you can use the formula for a geometric sequence. You will need to know the first term and the common ratio.

Q: What is the explicit formula for the sequence defined by the formula f(x+1)=23(f(x)f(x+1)=\frac{2}{3}(f(x), where f(1)=18f(1)=18?

A: The explicit formula for this sequence is f(x)=18⋅(23)(x−1)f(x) = 18 \cdot \left(\frac{2}{3}\right)^{(x-1)}.

Q: How do I verify that the explicit formula is correct?

A: To verify that the explicit formula is correct, you can plug in some values for xx and see if you get the correct values for f(x)f(x).

Q: What is the difference between an explicit formula and a recursive formula?

A: An explicit formula is a formula that gives the value of a term directly, while a recursive formula is a formula that gives the value of a term in terms of the previous term.

Q: Can I use a recursive formula to find the explicit formula for a geometric sequence?

A: Yes, you can use a recursive formula to find the explicit formula for a geometric sequence. However, it may be more difficult to find the explicit formula using a recursive formula.

Conclusion

In this article, we have answered some common questions about geometric sequences and explicit formulas. We have also provided examples and explanations to help you understand the concepts.

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References

  • [1] "Geometric Sequences". Math Open Reference. Retrieved 2023-02-26.
  • [2] "Geometric Sequences". Khan Academy. Retrieved 2023-02-26.

Frequently Asked Questions

  • 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 formula for a geometric sequence? A: The formula for a geometric sequence is an=a1â‹…r(n−1)a_n = a_1 \cdot r^{(n-1)}, where ana_n is the nth term of the sequence, a1a_1 is the first term, rr is the common ratio, and nn is the term number.
  • Q: How do I find the explicit formula for a geometric sequence? A: To find the explicit formula for a geometric sequence, you can use the formula for a geometric sequence. You will need to know the first term and the common ratio.
  • Q: What is the explicit formula for the sequence defined by the formula f(x+1)=23(f(x)f(x+1)=\frac{2}{3}(f(x), where f(1)=18f(1)=18? A: The explicit formula for this sequence is f(x)=18â‹…(23)(x−1)f(x) = 18 \cdot \left(\frac{2}{3}\right)^{(x-1)}.
  • Q: How do I verify that the explicit formula is correct? A: To verify that the explicit formula is correct, you can plug in some values for xx and see if you get the correct values for f(x)f(x).
  • Q: What is the difference between an explicit formula and a recursive formula? A: An explicit formula is a formula that gives the value of a term directly, while a recursive formula is a formula that gives the value of a term in terms of the previous term.
  • Q: Can I use a recursive formula to find the explicit formula for a geometric sequence? A: Yes, you can use a recursive formula to find the explicit formula for a geometric sequence. However, it may be more difficult to find the explicit formula using a recursive formula.