The Formula and the Sequence

Oct 13, 2025 by windailys.com 28 views

Introduction

Shanna has attempted to represent a given sequence using a formula. The sequence provided is $2, 5, 12.5, 31.25$ , and Shanna has used the formula $f(x+1)=2.5 f(x)$ when $f(1)=2$ to represent it. However, upon closer inspection, it appears that Shanna may have made an error in her representation. In this article, we will analyze Shanna's formula and identify the mistake she made.

The Formula and the Sequence

Shanna's formula is $f(x+1)=2.5 f(x)$ , and she has used the initial value $f(1)=2$ to represent the sequence. To understand the error, let's first examine the sequence and see if we can identify a pattern.

The given sequence is $2, 5, 12.5, 31.25$ . We can see that each term is obtained by multiplying the previous term by a certain factor. To find this factor, we can divide each term by its previous term.

$5/2 = 2.5$
$12.5/5 = 2.5$
$31.25/12.5 = 2.5$

We can see that the common ratio between consecutive terms is $2.5$ . This means that each term is obtained by multiplying the previous term by $2.5$ .

Analyzing Shanna's Formula

Now that we have identified the common ratio, let's analyze Shanna's formula. Shanna's formula is $f(x+1)=2.5 f(x)$ , which means that each term is obtained by multiplying the previous term by $2.5$ . However, Shanna has used the initial value $f(1)=2$ to represent the sequence.

To verify if Shanna's formula is correct, let's use it to generate the next term in the sequence. We can start with the initial value $f(1)=2$ and use the formula to find the next term.

$f(2) = 2.5 f(1) = 2.5 \times 2 = 5$

We can see that the first term generated by Shanna's formula is $5$ , which is correct. However, let's continue using the formula to generate the next term.

$f(3) = 2.5 f(2) = 2.5 \times 5 = 12.5$

We can see that the second term generated by Shanna's formula is $12.5$ , which is correct. However, let's continue using the formula to generate the next term.

$f(4) = 2.5 f(3) = 2.5 \times 12.5 = 31.25$

We can see that the third term generated by Shanna's formula is $31.25$ , which is correct. However, let's continue using the formula to generate the next term.

$f(5) = 2.5 f(4) = 2.5 \times 31.25 = 78.125$

We can see that the fourth term generated by Shanna's formula is $78.125$ , which is not correct. The correct fourth term in the sequence is $31.25$ , not $78.125$ .

Conclusion

In conclusion, Shanna's formula $f(x+1)=2.5 f(x)$ when $f(1)=2$ is not entirely correct. While the formula is correct in terms of the common ratio, the initial value used by Shanna is incorrect. The correct initial value should be $f(1)=1$ , not $f(1)=2$ . This is because the sequence starts with $2$ , which is obtained by multiplying the previous term by $2.5$ . Therefore, the correct formula should be $f(x+1)=2.5 f(x)$ when $f(1)=1$ .

The Correct Formula

The correct formula to represent the sequence $2, 5, 12.5, 31.25$ is $f(x+1)=2.5 f(x)$ when $f(1)=1$ . This formula is correct because it uses the correct common ratio and initial value to generate the sequence.

The Error Made by Shanna

The error made by Shanna is that she used the incorrect initial value. The correct initial value should be $f(1)=1$ , not $f(1)=2$ . This is because the sequence starts with $2$ , which is obtained by multiplying the previous term by $2.5$ . Therefore, the correct formula should be $f(x+1)=2.5 f(x)$ when $f(1)=1$ .

Conclusion

The Correct Answer

The correct answer is that Shanna used the incorrect initial value. The correct initial value should be $f(1)=1$ , not $f(1)=2$ . This is because the sequence starts with $2$ , which is obtained by multiplying the previous term by $2.5$ . Therefore, the correct formula should be $f(x+1)=2.5 f(x)$ when $f(1)=1$ .

The Final Answer

Q: What is the formula used by Shanna to represent the sequence?

A: Shanna's formula is $f(x+1)=2.5 f(x)$ when $f(1)=2$ .

Q: What is the sequence represented by Shanna's formula?

A: The sequence represented by Shanna's formula is $2, 5, 12.5, 31.25$ .

Q: What is the common ratio between consecutive terms in the sequence?

A: The common ratio between consecutive terms in the sequence is $2.5$ .

Q: Why is Shanna's formula not entirely correct?

A: Shanna's formula is not entirely correct because she used the incorrect initial value. The correct initial value should be $f(1)=1$ , not $f(1)=2$ .

Q: What is the correct formula to represent the sequence?

A: The correct formula to represent the sequence is $f(x+1)=2.5 f(x)$ when $f(1)=1$ .

Q: What is the error made by Shanna?

A: The error made by Shanna is that she used the incorrect initial value. The correct initial value should be $f(1)=1$ , not $f(1)=2$ .

Q: How can we verify if Shanna's formula is correct?

A: We can verify if Shanna's formula is correct by using it to generate the next term in the sequence. If the generated term matches the actual term in the sequence, then the formula is correct.

Q: What is the significance of the initial value in Shanna's formula?

A: The initial value in Shanna's formula is significant because it determines the starting point of the sequence. If the initial value is incorrect, then the entire sequence will be incorrect.

Q: How can we ensure that we use the correct initial value in a formula?

A: We can ensure that we use the correct initial value in a formula by carefully examining the sequence and identifying the first term. We can then use this first term as the initial value in the formula.

Q: What is the importance of using the correct formula to represent a sequence?

A: The importance of using the correct formula to represent a sequence is that it ensures that the sequence is accurately represented and can be used for further analysis and calculations.

Q: How can we use Shanna's formula to generate the next term in the sequence?

A: We can use Shanna's formula to generate the next term in the sequence by substituting the previous term into the formula and calculating the result.

Q: What is the next term in the sequence generated by Shanna's formula?

A: The next term in the sequence generated by Shanna's formula is $78.125$ , but this is incorrect. The correct next term in the sequence is not provided in the problem.

Q: How can we find the correct next term in the sequence?

A: We can find the correct next term in the sequence by using the correct formula and initial value. The correct formula is $f(x+1)=2.5 f(x)$ when $f(1)=1$ . We can use this formula to generate the next term in the sequence.

Q: What is the correct next term in the sequence?

A: The correct next term in the sequence is not provided in the problem, but we can find it by using the correct formula and initial value. The correct formula is $f(x+1)=2.5 f(x)$ when $f(1)=1$ . We can use this formula to generate the next term in the sequence.