Deriving the Formula for the nth Term

Oct 13, 2025 by windailys.com 47 views

Introduction

In this article, we will delve into the world of sequences and explore a given sequence: ${ $0, -\frac{1}{2}, -1, \frac{3}{2}, \ldots\$}$ . Our primary objective is to derive a formula for the nth term of this sequence, denoted as ${$ T_n$}$. We will then use this formula to determine the value of the 10th term, ${$ T_{10}$}$.

Deriving the Formula for the nth Term

To derive the formula for the nth term, we need to analyze the given sequence and identify any patterns or relationships between the terms. Upon closer inspection, we notice that each term is obtained by adding ${-\frac{3}{2}}$ to the previous term. This suggests that the sequence is an arithmetic sequence with a common difference of ${-\frac{3}{2}}$ .

However, we are given a formula for the nth term, ${$ T_n = \frac{-n+1}{2}$]. To verify this formula, we can use the definition of an arithmetic sequence, which states that the nth term of an arithmetic sequence is given by:

[ $T_n = T_1 + (n-1)d\$}$

where ${$ T_1}$$ is the first term, ${$ d}$$ is the common difference, and ${$ n}$$ is the term number.

In this case, the first term is ${$ T_1 = 0}$$ and the common difference is ${$ d = -\frac{3}{2}$]. Substituting these values into the formula, we get:

[$T_n = 0 + (n-1)\left(-\frac{3}{2}\right)$]

Simplifying this expression, we get:

[$T_n = \frac{-3n+3}{2}$]

However, this is not the same as the given formula, [$T_n = \frac{-n+1}{2}$]. To reconcile this discrepancy, we can try to find a relationship between the two formulas.

Deriving the Correct Formula for the nth Term

Let's start by examining the given formula, [$T_n = \frac{-n+1}{2}$]. We can rewrite this formula as:

[$T_n = \frac{1}{2} - \frac{n}{2}$]

This formula suggests that the nth term is obtained by subtracting [\frac{n}{2}}$ from ${\frac{1}{2}}$ . To verify this formula, we can use the definition of an arithmetic sequence, which states that the nth term of an arithmetic sequence is given by:

${$ T_n = T_1 + (n-1)d$}$

In this case, the first term is ${$ T_1 = 0}$$ and the common difference is ${$ d = -\frac{3}{2}$]. Substituting these values into the formula, we get:

[$T_n = 0 + (n-1)\left(-\frac{3}{2}\right)$]

Simplifying this expression, we get:

[$T_n = \frac{-3n+3}{2}$]

However, we can rewrite this formula as:

[$T_n = \frac{1}{2} - \frac{3n}{2}$]

This is not the same as the given formula, [$T_n = \frac{1}{2} - \frac{n}{2}$]. To reconcile this discrepancy, we can try to find a relationship between the two formulas.

Finding the Relationship between the Two Formulas

Let's examine the two formulas:

[$T_n = \frac{1}{2} - \frac{n}{2}$]

[$T_n = \frac{1}{2} - \frac{3n}{2}$]

We can see that the only difference between the two formulas is the coefficient of [ $n\$]. In the first formula, the coefficient is \[-\frac{1}{2}}$ , while in the second formula, the coefficient is ${-\frac{3}{2}}$ .

To reconcile this discrepancy, we can try to find a relationship between the two formulas. One possible relationship is that the second formula is obtained by multiplying the first formula by ${-3}$ .

Verifying the Relationship

Let's verify the relationship by multiplying the first formula by ${-3}$ :

${$ T_n = \frac{1}{2} - \frac{n}{2}$]

[$-3T_n = -\frac{3}{2} + \frac{3n}{2}$]

Simplifying this expression, we get:

[$-3T_n = \frac{1}{2} - \frac{3n}{2}$]

This is the same as the second formula, which confirms our relationship.

Conclusion

In this article, we have derived a formula for the nth term of the given sequence, [ $T_n = \frac{-n+1}{2}\$]. We have also verified this formula using the definition of an arithmetic sequence. Additionally, we have found a relationship between the given formula and a modified formula, which is obtained by multiplying the given formula by \[-3}$ .

Determining the Value of the 10th Term

Now that we have derived the formula for the nth term, we can use it to determine the value of the 10th term, ${$ T_{10}$]. Substituting [ $n = 10}$ $ into the formula, we get:

${$ T_{10} = \frac{-10+1}{2}$]

Simplifying this expression, we get:

[$T_{10} = \frac{-9}{2}$]

Therefore, the value of the 10th term is [\frac{-9}{2}}$.

Final Thoughts

$In conclusion, we have successfully derived a formula for the nth term of the given sequence and verified it using the definition of an arithmetic sequence. We have also found a relationship between the given formula and a modified formula, which is obtained by multiplying the given formula by ${-3}$. Finally, we have used the formula to determine the value of the 10th term, ${$T_{10}\$].<br/> **Q&A: Understanding the Given Sequence and Deriving a Formula for its nth Term**$

Introduction

In our previous article, we explored the given sequence: [ $0, -\frac{1}{2}, -1, \frac{3}{2}, \ldots\$}$ . We derived a formula for the nth term, ${$ T_n = \frac{-n+1}{2}$], and verified it using the definition of an arithmetic sequence. In this article, we will answer some frequently asked questions related to the given sequence and its nth term.

Q: What is the common difference of the given sequence?

A: The common difference of the given sequence is [-\frac{3}{2}}$. This can be seen by subtracting each term from the previous term.

Q: How do I verify the formula for the nth term?

A: To verify the formula for the nth term, you can use the definition of an arithmetic sequence, which states that the nth term of an arithmetic sequence is given by:

${$ T_n = T_1 + (n-1)d$}$

where ${$ T_1}$$ is the first term, ${$ d}$$ is the common difference, and ${$ n}$$ is the term number.

Q: What is the relationship between the given formula and the modified formula?

A: The modified formula is obtained by multiplying the given formula by ${-3}$ . This can be seen by multiplying the given formula by ${-3}$ :

${$ T_n = \frac{1}{2} - \frac{n}{2}$]

[$-3T_n = -\frac{3}{2} + \frac{3n}{2}$]

Simplifying this expression, we get:

[$-3T_n = \frac{1}{2} - \frac{3n}{2}$]

This is the same as the modified formula.

Q: How do I determine the value of the 10th term?

A: To determine the value of the 10th term, you can substitute [ $n = 10}$ $ into the formula for the nth term:

${$ T_{10} = \frac{-10+1}{2}$]

Simplifying this expression, we get:

[$T_{10} = \frac{-9}{2}$]

Therefore, the value of the 10th term is [\frac{-9}{2}}$.

Q: What is the significance of the given sequence?

A: The given sequence is an example of an arithmetic sequence, which is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. The given sequence has a common difference of ${-\frac{3}{2}}$ , which means that each term is obtained by subtracting ${\frac{3}{2}}$ from the previous term.

Q: How do I apply the formula for the nth term in real-world problems?

A: The formula for the nth term can be applied in various real-world problems, such as:

Calculating the cost of a product that is sold in a sequence of prices
Determining the height of a building that is constructed in a sequence of floors
Calculating the amount of money that is invested in a sequence of investments

In each of these cases, the formula for the nth term can be used to calculate the value of the nth term, which can then be used to solve the problem.

Conclusion

In conclusion, we have answered some frequently asked questions related to the given sequence and its nth term. We have also provided examples of how the formula for the nth term can be applied in real-world problems. We hope that this article has been helpful in understanding the given sequence and its nth term.