Determining the nth Term of an Arithmetic Sequence

Oct 13, 2025 by miodeya.com 149 views

Introduction

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this article, we will explore the first three terms of an arithmetic sequence, which are -1, 2, and 5.50. We will determine the nth term, Tn, of the sequence and calculate specific terms, including T43.

Determining the nth Term of an Arithmetic Sequence

To determine the nth term of an arithmetic sequence, we need to find the common difference (d) between consecutive terms. The common difference is the constant difference between any two consecutive terms in the sequence.

Finding the Common Difference

The common difference (d) can be found by subtracting any term from the previous term. In this case, we can subtract the first term (-1) from the second term (2) to find the common difference.

d = 2 - (-1) d = 2 + 1 d = 3

Determining the nth Term

Now that we have found the common difference (d), we can use the formula for the nth term of an arithmetic sequence to determine the nth term, Tn.

Tn = a + (n - 1)d

where:

Tn is the nth term
a is the first term
n is the term number
d is the common difference

In this case, the first term (a) is -1, the common difference (d) is 3, and we want to find the nth term. Plugging in these values, we get:

Tn = -1 + (n - 1)(3)

Simplifying the Formula

We can simplify the formula by distributing the 3 to the (n - 1) term.

Tn = -1 + 3n - 3

Combine like terms:

Tn = 3n - 4

Conclusion

We have determined the formula for the nth term of the arithmetic sequence, which is Tn = 3n - 4.

Calculating T43

Now that we have determined the formula for the nth term, we can calculate T43 by plugging in n = 43 into the formula.

T43 = 3(43) - 4 T43 = 129 - 4 T43 = 125

Conclusion

We have calculated T43, which is 125.

Evaluating the Sequence

Now that we have determined the formula for the nth term and calculated T43, we can evaluate the sequence.

Evaluating the First Few Terms

Let's evaluate the first few terms of the sequence using the formula Tn = 3n - 4.

T1 = 3(1) - 4 T1 = 3 - 4 T1 = -1

T2 = 3(2) - 4 T2 = 6 - 4 T2 = 2

T3 = 3(3) - 4 T3 = 9 - 4 T3 = 5

Conclusion

We have evaluated the first few terms of the sequence, which are -1, 2, and 5.

Conclusion

In this article, we have determined the formula for the nth term of an arithmetic sequence, which is Tn = 3n - 4. We have also calculated T43, which is 125. Finally, we have evaluated the sequence by calculating the first few terms.

References

Glossary

Arithmetic Sequence: A sequence of numbers in which the difference between any two consecutive terms is constant.
Common Difference: The constant difference between any two consecutive terms in an arithmetic sequence.
Nth Term: The term in an arithmetic sequence that corresponds to a given term number (n).
Term Number: The position of a term in an arithmetic sequence, starting from 1.
The First Three Terms of an Arithmetic Sequence: Determining the nth Term and Calculating Specific Terms ===========================================================

Q&A: Frequently Asked Questions

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.

Q: How do I find the common difference (d) of an arithmetic sequence?

A: To find the common difference (d), subtract any term from the previous term. For example, if the first term is -1 and the second term is 2, the common difference (d) is 2 - (-1) = 3.

Q: What is the formula for the nth term of an arithmetic sequence?

A: The formula for the nth term of an arithmetic sequence is Tn = a + (n - 1)d, where:

Tn is the nth term
a is the first term
n is the term number
d is the common difference

Q: How do I determine the nth term of an arithmetic sequence?

A: To determine the nth term, plug in the values of a, n, and d into the formula Tn = a + (n - 1)d.

Q: Can I use the formula to calculate any term in an arithmetic sequence?

A: Yes, you can use the formula to calculate any term in an arithmetic sequence by plugging in the values of a, n, and d.

Q: What is the significance of the common difference (d) in an arithmetic sequence?

A: The common difference (d) is the constant difference between any two consecutive terms in an arithmetic sequence. It is used to determine the nth term of the sequence.

Q: Can I use the formula to calculate the first term of an arithmetic sequence?

A: No, the formula Tn = a + (n - 1)d is used to calculate the nth term, not the first term. To find the first term, you need to know the value of the nth term and the common difference (d).

Q: How do I evaluate an arithmetic sequence?

A: To evaluate an arithmetic sequence, calculate the first few terms using the formula Tn = a + (n - 1)d.

Q: Can I use the formula to calculate the sum of an arithmetic sequence?

A: No, the formula Tn = a + (n - 1)d is used to calculate the nth term, not the sum of the sequence. To find the sum of an arithmetic sequence, you need to use a different formula.

Q: What is the relationship between the common difference (d) and the nth term of an arithmetic sequence?

A: The common difference (d) is used to determine the nth term of an arithmetic sequence. The formula Tn = a + (n - 1)d shows that the nth term is equal to the first term (a) plus the product of the common difference (d) and the term number (n - 1).

Q: Can I use the formula to calculate the term number (n) of an arithmetic sequence?

A: No, the formula Tn = a + (n - 1)d is used to calculate the nth term, not the term number (n). To find the term number (n), you need to know the value of the nth term and the common difference (d).

Conclusion

In this Q&A article, we have answered frequently asked questions about arithmetic sequences, including how to find the common difference (d), determine the nth term, and evaluate the sequence. We have also discussed the significance of the common difference (d) and the relationship between the common difference (d) and the nth term of an arithmetic sequence.

References

Glossary

Arithmetic Sequence: A sequence of numbers in which the difference between any two consecutive terms is constant.
Common Difference: The constant difference between any two consecutive terms in an arithmetic sequence.
Nth Term: The term in an arithmetic sequence that corresponds to a given term number (n).
Term Number: The position of a term in an arithmetic sequence, starting from 1.