The $n$th Term Of A Sequence Is $3n^2 - 1$.a) Find The Second Term Of The Sequence. $\square$b) Find The Fifth Term Of The Sequence. $\square$

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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 a value of $n$ into the formula. In this article, we will explore the $n$th term of a sequence given by the formula $3n^2 - 1$. We will use this formula to find the second and fifth terms of the sequence.

The Formula for the $n$th Term

The formula for the $n$th term of the sequence is given by:

$$a_n = 3n^2 - 1$$

where $a_n$ represents the $n$th term of the sequence.

Finding the Second Term

To find the second term of the sequence, we need to substitute $n = 2$ into the formula.

$$a_2 = 3(2)^2 - 1$$

$$a_2 = 3(4) - 1$$

$$a_2 = 12 - 1$$

$$a_2 = 11$$

Therefore, the second term of the sequence is $11$.

Finding the Fifth Term

To find the fifth term of the sequence, we need to substitute $n = 5$ into the formula.

$$a_5 = 3(5)^2 - 1$$

$$a_5 = 3(25) - 1$$

$$a_5 = 75 - 1$$

$$a_5 = 74$$

Therefore, the fifth term of the sequence is $74$.

Discussion

Sequences are an essential concept in mathematics, and understanding how to find specific terms of a sequence is crucial. The formula for the $n$th term of a sequence can be used to find any term in the sequence by substituting the corresponding value of $n$ into the formula. In this article, we used the formula $3n^2 - 1$ to find the second and fifth terms of the sequence.

Conclusion

In conclusion, finding specific terms of a sequence is a fundamental concept in mathematics. By using the formula for the $n$th term of a sequence, we can find any term in the sequence. In this article, we used the formula $3n^2 - 1$ to find the second and fifth terms of the sequence. We hope that this article has provided a clear understanding of how to find specific terms of a sequence.

Example Problems

Problem 1

Find the third term of the sequence given by the formula $2n^2 + 1$.

Solution

To find the third term of the sequence, we need to substitute $n = 3$ into the formula.

$$a_3 = 2(3)^2 + 1$$

$$a_3 = 2(9) + 1$$

$$a_3 = 18 + 1$$

$$a_3 = 19$$

Therefore, the third term of the sequence is $19$.

Problem 2

Find the fourth term of the sequence given by the formula $n^2 - 2$.

Solution

To find the fourth term of the sequence, we need to substitute $n = 4$ into the formula.

$$a_4 = (4)^2 - 2$$

$$a_4 = 16 - 2$$

$$a_4 = 14$$

Therefore, the fourth term of the sequence is $14$.

Applications of Sequences

Sequences have numerous applications in mathematics, science, and engineering. Some of the applications of sequences include:

• Mathematics: Sequences are used to describe the behavior of mathematical functions and to solve mathematical problems.
• Science: Sequences are used to model the behavior of physical systems, such as population growth and chemical reactions.
• Engineering: Sequences are used to design and optimize systems, such as control systems and signal processing systems.

Conclusion

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Introduction

In our previous article, we explored the $n$th term of a sequence given by the formula $3n^2 - 1$. We used this formula to find the second and fifth terms of the sequence. In this article, we will answer some frequently asked questions about sequences and the $n$th 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 a value of $n$ into the formula for the $n$th term.

Q: How do I find the $n$th term of a sequence?

A: To find the $n$th term of a sequence, you need to substitute $n$ into the formula for the $n$th term.

Q: What is the formula for the $n$th term of a sequence?

A: The formula for the $n$th term of a sequence is given by:

$$a_n = 3n^2 - 1$$

where $a_n$ represents the $n$th term of the sequence.

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

A: To find the second term of a sequence, you need to substitute $n = 2$ into the formula for the $n$th term.

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

A: To find the fifth term of a sequence, you need to substitute $n = 5$ into the formula for the $n$th term.

Q: What are some common applications of sequences?

A: Sequences have numerous applications in mathematics, science, and engineering. Some of the applications of sequences include:

• Mathematics: Sequences are used to describe the behavior of mathematical functions and to solve mathematical problems.
• Science: Sequences are used to model the behavior of physical systems, such as population growth and chemical reactions.
• Engineering: Sequences are used to design and optimize systems, such as control systems and signal processing systems.

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.

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

A: To find the product of a sequence, you need to multiply all the terms in the sequence.

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 a sequence.

Q: How do I find the $n$th term of a sequence with a variable exponent?

A: To find the $n$th term of a sequence with a variable exponent, you need to substitute $n$ into the formula for the $n$th term, taking into account the variable exponent.

Q: How do I find the $n$th term of a sequence with a variable coefficient?

A: To find the $n$th term of a sequence with a variable coefficient, you need to substitute $n$ into the formula for the $n$th term, taking into account the variable coefficient.

Conclusion

In conclusion, sequences are an essential concept in mathematics, and understanding how to find specific terms of a sequence is crucial. By using the formula for the $n$th term of a sequence, we can find any term in the sequence. In this article, we answered some frequently asked questions about sequences and the $n$th term of a sequence. We hope that this article has provided a clear understanding of how to find specific terms of a sequence.

Example Problems

Problem 1

Find the third term of the sequence given by the formula $2n^2 + 1$.

Solution

To find the third term of the sequence, we need to substitute $n = 3$ into the formula.

$$a_3 = 2(3)^2 + 1$$

$$a_3 = 2(9) + 1$$

$$a_3 = 18 + 1$$

$$a_3 = 19$$

Therefore, the third term of the sequence is $19$.

Problem 2

Find the fourth term of the sequence given by the formula $n^2 - 2$.

Solution

To find the fourth term of the sequence, we need to substitute $n = 4$ into the formula.

$$a_4 = (4)^2 - 2$$

$$a_4 = 16 - 2$$

$$a_4 = 14$$

Therefore, the fourth term of the sequence is $14$.

Applications of Sequences

Sequences have numerous applications in mathematics, science, and engineering. Some of the applications of sequences include:

• Mathematics: Sequences are used to describe the behavior of mathematical functions and to solve mathematical problems.
• Science: Sequences are used to model the behavior of physical systems, such as population growth and chemical reactions.
• Engineering: Sequences are used to design and optimize systems, such as control systems and signal processing systems.

Conclusion

In conclusion, sequences are an essential concept in mathematics, and understanding how to find specific terms of a sequence is crucial. By using the formula for the $n$th term of a sequence, we can find any term in the sequence. In this article, we answered some frequently asked questions about sequences and the $n$th term of a sequence. We hope that this article has provided a clear understanding of how to find specific terms of a sequence.