Use formulas to find the nth term of a sequence

Oct 13, 2025 by miodeya.com 42 views

Introduction

In mathematics, sequences are an essential concept used to describe a series of numbers in a specific order. A sequence can be defined by a formula, and each term in the sequence is obtained by substituting the corresponding value of the index (or term number) into the formula. In this article, we will explore the nth term of a sequence given by the formula $3n^2 - 1$ and find the second and fifth terms of the sequence.

Understanding the Formula

The given formula is $3n^2 - 1$ , where $n$ represents the term number. To find a specific term in the sequence, we need to substitute the corresponding value of $n$ into the formula. For example, to find the second term, we will substitute $n = 2$ into the formula.

Finding the Second Term

To find the second term of the sequence, we will substitute $n = 2$ into the formula $3n^2 - 1$ .

\begin{aligned} \text{Second term} &= 3(2)^2 - 1 \\ &= 3(4) - 1 \\ &= 12 - 1 \\ &= 11 \end{aligned}

Therefore, the second term of the sequence is $\boxed{11}$ .

Finding the Fifth Term

To find the fifth term of the sequence, we will substitute $n = 5$ into the formula $3n^2 - 1$ .

\begin{aligned} \text{Fifth term} &= 3(5)^2 - 1 \\ &= 3(25) - 1 \\ &= 75 - 1 \\ &= 74 \end{aligned}

Therefore, the fifth term of the sequence is $\boxed{74}$ .

Discussion

Sequences are an essential concept in mathematics, and understanding how to find specific terms in a sequence is crucial. The formula $3n^2 - 1$ represents a quadratic sequence, where each term is obtained by squaring the term number and multiplying it by 3, then subtracting 1. By substituting the corresponding value of $n$ into the formula, we can find any term in the sequence.

Conclusion

In conclusion, we have found the second and fifth terms of the sequence given by the formula $3n^2 - 1$ . By understanding the formula and substituting the corresponding value of $n$ , we can find any term in the sequence. This concept is essential in mathematics and has numerous applications in various fields, including science, engineering, and economics.

Applications of Sequences

Sequences have numerous applications in various fields, including:

Science: Sequences are used to model population growth, chemical reactions, and other scientific phenomena.
Engineering: Sequences are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Sequences are used to model economic systems, including population growth and economic forecasting.

Real-World Examples

Sequences are used in various real-world applications, including:

Finance: Sequences are used to model stock prices and predict future stock prices.
Biology: Sequences are used to model population growth and predict the spread of diseases.
Computer Science: Sequences are used to design and optimize algorithms, such as sorting and searching algorithms.

Conclusion

Introduction

In our previous article, we explored the nth term of a sequence given by the formula $3n^2 - 1$ and found the second and fifth terms of the sequence. In this article, we will answer some frequently asked questions about sequences and the nth term of a sequence.

Q&A

Q: What is a sequence?

A: A sequence is a series of numbers in a specific order. Each term in the sequence is obtained by substituting the corresponding value of the index (or term number) into a formula.

Q: What is the nth term of a sequence?

A: The nth term of a sequence is the term in the sequence that corresponds to the nth index. It is obtained by substituting the value of n into the formula that defines the sequence.

Q: How do I find the nth term of a sequence?

A: To find the nth term of a sequence, you need to substitute the value of n into the formula that defines the sequence. For example, if the formula is $3n^2 - 1$ , you would substitute n = 2 to find the second term.

Q: What is the difference between a sequence and a series?

A: A sequence is a series of numbers in a specific order, while a series is the sum of the terms in a sequence.

Q: How do I find the sum of a sequence?

A: To find the sum of a sequence, you need to add up all the terms in the sequence. This can be done using various methods, including the formula for the sum of a geometric series.

Q: What is a geometric sequence?

A: A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed constant.

Q: How do I find the nth term of a geometric sequence?

A: To find the nth term of a geometric sequence, you need to substitute the value of n into the formula that defines the sequence. For example, if the formula is $2^n$ , you would substitute n = 3 to find the fourth term.

Q: What is the formula for the sum of a geometric series?

A: The formula for the sum of a geometric series is $S_n = \frac{a(1 - r^n)}{1 - r}$ , where a is the first term, r is the common ratio, and n is the number of terms.

Q: How do I use the formula for the sum of a geometric series?

A: To use the formula for the sum of a geometric series, you need to substitute the values of a, r, and n into the formula. For example, if a = 2, r = 3, and n = 4, you would substitute these values into the formula to find the sum of the first 4 terms of the sequence.

Conclusion

Real-World Examples

Sequences and the nth term of a sequence have numerous real-world applications, including:

Finance: Sequences are used to model stock prices and predict future stock prices.
Biology: Sequences are used to model population growth and predict the spread of diseases.
Computer Science: Sequences are used to design and optimize algorithms, such as sorting and searching algorithms.

Tips and Tricks

Use formulas to find the nth term of a sequence: Formulas are a quick and easy way to find the nth term of a sequence.
Use the formula for the sum of a geometric series: The formula for the sum of a geometric series is a powerful tool for finding the sum of a sequence.
Practice, practice, practice: The more you practice finding the nth term of a sequence and using formulas, the more comfortable you will become with these skills.

Conclusion

In conclusion, sequences and the nth term of a sequence are essential concepts in mathematics. Understanding how to find specific terms in a sequence and how to use formulas to find the sum of a sequence are crucial skills that have numerous applications in various fields, including science, engineering, and economics. By following the tips and tricks outlined in this article, you will be well on your way to becoming proficient in finding the nth term of a sequence and using formulas to find the sum of a sequence.