The Function and Its Application

Oct 13, 2025 by windailys.com 113 views

In mathematics, a sequence is a series of numbers in a specific order, often defined by a mathematical function. The given sequence is represented by the function $3n + 2$ , where $n$ is the term number. To determine the term value for the term number 9, we need to understand how the function works and apply it to find the value of the 9th term.

The Function and Its Application

The function $3n + 2$ takes the term number $n$ as input and returns the corresponding term value. To find the term value for a given term number, we simply substitute the term number into the function and evaluate it. For example, to find the term value for the 1st term, we substitute $n = 1$ into the function:

3(1) + 2 = 3 + 2 = 5

This means that the term value for the 1st term is 5.

Calculating the Term Value for the 9th Term

To find the term value for the 9th term, we substitute $n = 9$ into the function:

3(9) + 2 = 27 + 2 = 29

Therefore, the term value for the 9th term is 29.

Understanding the Pattern and Formula

The given sequence is a simple arithmetic sequence, where each term is obtained by adding a fixed constant to the previous term. In this case, the common difference between consecutive terms is 2. We can use this pattern to find the term value for any term number.

Deriving the Formula

Let's derive the formula for the term value in terms of the term number. We know that the term value for the 1st term is 3, and the common difference between consecutive terms is 2. We can use this information to find the term value for any term number $n$ :

\text{Term Value} = 3 + (n - 1) \times 2

Simplifying the formula, we get:

\text{Term Value} = 3 + 2n - 2

\text{Term Value} = 2n + 1

This is the formula for the term value in terms of the term number.

Applying the Formula

We can use the formula to find the term value for any term number. For example, to find the term value for the 9th term, we substitute $n = 9$ into the formula:

\text{Term Value} = 2(9) + 1

\text{Term Value} = 18 + 1

\text{Term Value} = 19

However, we previously found that the term value for the 9th term is 29. This discrepancy arises from the fact that the formula we derived is not accurate for this sequence.

Revisiting the Sequence and Function

Let's revisit the sequence and function to understand why the formula we derived is not accurate. The sequence is represented by the function $3n + 2$ , where $n$ is the term number. To find the term value for a given term number, we substitute the term number into the function and evaluate it.

Re-Deriving the Formula

Let's re-derive the formula for the term value in terms of the term number. We know that the term value for the 1st term is 3, and the common difference between consecutive terms is 2. We can use this information to find the term value for any term number $n$ :

\text{Term Value} = 3 + (n - 1) \times 3

Simplifying the formula, we get:

\text{Term Value} = 3 + 3n - 3

\text{Term Value} = 3n

This is the correct formula for the term value in terms of the term number.

Applying the Correct Formula

We can use the correct formula to find the term value for any term number. For example, to find the term value for the 9th term, we substitute $n = 9$ into the formula:

\text{Term Value} = 3(9)

\text{Term Value} = 27

However, we previously found that the term value for the 9th term is 29. This discrepancy arises from the fact that the correct formula we derived is not accurate for this sequence.

Revisiting the Sequence and Function Again

Let's revisit the sequence and function to understand why the correct formula we derived is not accurate. The sequence is represented by the function $3n + 2$ , where $n$ is the term number. To find the term value for a given term number, we substitute the term number into the function and evaluate it.

Re-Deriving the Formula Again

\text{Term Value} = 3n + 2

This is the correct formula for the term value in terms of the term number.

Applying the Correct Formula Again

We can use the correct formula to find the term value for any term number. For example, to find the term value for the 9th term, we substitute $n = 9$ into the formula:

\text{Term Value} = 3(9) + 2

\text{Term Value} = 27 + 2

\text{Term Value} = 29

This time, we get the correct answer, which is 29.

Conclusion

In this article, we will answer some frequently asked questions related to the sequence and function $3n + 2$ . We will cover topics such as the formula for the term value, the common difference between consecutive terms, and how to apply the formula to find the term value for any term number.

Q: What is the formula for the term value in terms of the term number?

A: The formula for the term value in terms of the term number is $3n + 2$ , where $n$ is the term number.

Q: What is the common difference between consecutive terms?

A: The common difference between consecutive terms is 2.

Q: How do I apply the formula to find the term value for any term number?

A: To apply the formula, simply substitute the term number into the formula and evaluate it. For example, to find the term value for the 9th term, you would substitute $n = 9$ into the formula:

\text{Term Value} = 3(9) + 2

\text{Term Value} = 27 + 2

\text{Term Value} = 29

Q: What if I want to find the term value for a term number that is not a multiple of 3?

A: If you want to find the term value for a term number that is not a multiple of 3, you can still apply the formula. For example, to find the term value for the 5th term, you would substitute $n = 5$ into the formula:

\text{Term Value} = 3(5) + 2

\text{Term Value} = 15 + 2

\text{Term Value} = 17

Q: Can I use the formula to find the term value for a negative term number?

A: No, you cannot use the formula to find the term value for a negative term number. The formula is only valid for positive term numbers.

Q: What if I want to find the term value for a term number that is greater than the number of terms in the sequence?

A: If you want to find the term value for a term number that is greater than the number of terms in the sequence, you will get an undefined result. For example, if the sequence only has 10 terms, you cannot find the term value for the 11th term.

Q: Can I use the formula to find the term value for a term number that is a fraction?

A: No, you cannot use the formula to find the term value for a term number that is a fraction. The formula is only valid for integer term numbers.

Conclusion

In this article, we answered some frequently asked questions related to the sequence and function $3n + 2$ . We covered topics such as the formula for the term value, the common difference between consecutive terms, and how to apply the formula to find the term value for any term number. We hope that this article has been helpful in answering your questions and providing a better understanding of the sequence and function.

Additional Resources

If you have any further questions or need additional help, please refer to the following resources:

The original article on the sequence and function $3n + 2$
The formula for the term value in terms of the term number
The common difference between consecutive terms
How to apply the formula to find the term value for any term number

We hope that this article has been helpful in providing a better understanding of the sequence and function $3n + 2$ . If you have any further questions or need additional help, please don't hesitate to ask.