Given The Sequence:$\[ -2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots \\]Which Formula Can Be Used To Describe The Sequence?A. \[$ F(x+1) = -2 F(x) \$\] B. \[$ F(x+1) = -\frac{1}{2} F(x)

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Uncovering the Hidden Pattern in a Sequence: A Mathematical Exploration

In the world of mathematics, sequences are an essential concept that helps us understand patterns and relationships between numbers. A sequence is a list of numbers in a specific order, and it can be described using various formulas. In this article, we will delve into a given sequence and explore the formula that can be used to describe it.

The given sequence is:

−223,−513,−1023,−2113,−4223,…{ -2 \frac{2}{3}, -5 \frac{1}{3}, -10 \frac{2}{3}, -21 \frac{1}{3}, -42 \frac{2}{3}, \ldots }

At first glance, this sequence appears to be a list of random numbers. However, upon closer inspection, we can notice a pattern emerging. The numbers seem to be increasing in magnitude, but with a twist. The sequence is not just a simple arithmetic progression, but rather a more complex pattern.

To understand the sequence better, let's examine the differences between consecutive terms.

  • The difference between the first and second term is: −513−(−223)=−213-5 \frac{1}{3} - (-2 \frac{2}{3}) = -2 \frac{1}{3}
  • The difference between the second and third term is: −1023−(−513)=−513-10 \frac{2}{3} - (-5 \frac{1}{3}) = -5 \frac{1}{3}
  • The difference between the third and fourth term is: −2113−(−1023)=−1023-21 \frac{1}{3} - (-10 \frac{2}{3}) = -10 \frac{2}{3}
  • The difference between the fourth and fifth term is: −4223−(−2113)=−2113-42 \frac{2}{3} - (-21 \frac{1}{3}) = -21 \frac{1}{3}

As we can see, the differences between consecutive terms are not constant, but rather they are increasing in magnitude. This suggests that the sequence is not a simple arithmetic progression, but rather a more complex pattern.

Now that we have analyzed the sequence, let's explore the formula that can be used to describe it. The formula is given as:

f(x+1)=−2f(x){ f(x+1) = -2 f(x) }

This formula suggests that each term in the sequence is obtained by multiplying the previous term by -2. Let's test this formula by applying it to the given sequence.

  • The first term is: f(1)=−223f(1) = -2 \frac{2}{3}
  • The second term is: f(2)=−2f(1)=−2(−223)=−513f(2) = -2 f(1) = -2 (-2 \frac{2}{3}) = -5 \frac{1}{3}
  • The third term is: f(3)=−2f(2)=−2(−513)=−1023f(3) = -2 f(2) = -2 (-5 \frac{1}{3}) = -10 \frac{2}{3}
  • The fourth term is: f(4)=−2f(3)=−2(−1023)=−2113f(4) = -2 f(3) = -2 (-10 \frac{2}{3}) = -21 \frac{1}{3}
  • The fifth term is: f(5)=−2f(4)=−2(−2113)=−4223f(5) = -2 f(4) = -2 (-21 \frac{1}{3}) = -42 \frac{2}{3}

As we can see, the formula f(x+1)=−2f(x)f(x+1) = -2 f(x) accurately describes the given sequence.

In conclusion, the given sequence can be described using the formula f(x+1)=−2f(x)f(x+1) = -2 f(x). This formula suggests that each term in the sequence is obtained by multiplying the previous term by -2. By analyzing the sequence and exploring the formula, we have uncovered the hidden pattern in the sequence.

The given sequence is a classic example of a geometric progression, where each term is obtained by multiplying the previous term by a fixed constant. In this case, the constant is -2. The formula f(x+1)=−2f(x)f(x+1) = -2 f(x) is a simple and elegant way to describe the sequence.

However, it's worth noting that the sequence can also be described using another formula:

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

This formula suggests that each term in the sequence is obtained by multiplying the previous term by -1/2. While this formula is also correct, it's not as intuitive as the first formula, which suggests that each term is obtained by multiplying the previous term by -2.

In conclusion, the given sequence is a fascinating example of a geometric progression, and the formula f(x+1)=−2f(x)f(x+1) = -2 f(x) is a simple and elegant way to describe it. By analyzing the sequence and exploring the formula, we have uncovered the hidden pattern in the sequence. We hope that this article has provided a deeper understanding of the sequence and the formula that describes it.