Write A Rule For The $n$th Term Of The Arithmetic Sequence:${ -3, 4, 11, 18, \ldots }$

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this article, we will derive the rule for the $n$th term of the given arithmetic sequence: $-3, 4, 11, 18, \ldots$

Understanding the Arithmetic Sequence

The given arithmetic sequence is: $-3, 4, 11, 18, \ldots$ To derive the rule for the $n$th term, we need to understand the pattern of the sequence. Let's examine the differences between consecutive terms:

• $$4 - (-3) = 7$$

• $$11 - 4 = 7$$

• $$18 - 11 = 7$$

As we can see, the difference between consecutive terms is constant, which is $7$. This confirms that the given sequence is an arithmetic sequence.

Deriving the Rule for the $n$th Term

To derive the rule for the $n$th term, we need to find the first term and the common difference. The first term is $-3$, and the common difference is $7$.

The general formula for the $n$th term of an arithmetic sequence is:

$$a_n = a_1 + (n - 1)d$$

where $a_n$ is the $n$th term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Substituting the values, we get:

$$a_n = -3 + (n - 1)7$$

Simplifying the equation, we get:

$$a_n = -3 + 7n - 7$$

$$a_n = 7n - 10$$

Therefore, the rule for the $n$th term of the given arithmetic sequence is:

$$a_n = 7n - 10$$

Example

Let's find the 5th term of the sequence using the derived rule:

$$a_5 = 7(5) - 10$$

$$a_5 = 35 - 10$$

$$a_5 = 25$$

Therefore, the 5th term of the sequence is $25$.

Conclusion

In this article, we derived the rule for the $n$th term of the given arithmetic sequence: $-3, 4, 11, 18, \ldots$ We found that the first term is $-3$ and the common difference is $7$. Using the general formula for the $n$th term of an arithmetic sequence, we derived the rule:

$$a_n = 7n - 10$$

We also provided an example to demonstrate the use of the derived rule.

Applications of Arithmetic Sequences

Arithmetic sequences have numerous applications in various fields, including:

• Finance: Arithmetic sequences are used to calculate interest rates, investments, and annuities.
• Science: Arithmetic sequences are used to model population growth, chemical reactions, and physical phenomena.
• Engineering: Arithmetic sequences are used to design and optimize systems, such as electrical circuits and mechanical systems.

Real-World Examples

Arithmetic sequences are used in various real-world scenarios, including:

• Compound Interest: Arithmetic sequences are used to calculate compound interest rates, which are used to determine the future value of investments.
• Population Growth: Arithmetic sequences are used to model population growth, which is essential for urban planning and resource allocation.
• Music: Arithmetic sequences are used to create musical patterns and rhythms.

Final Thoughts

In this article, we will answer some frequently asked questions about arithmetic sequences, including their properties, applications, and examples.

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: What are the properties of an arithmetic sequence?

A: The properties of an arithmetic sequence include:

• Constant difference: The difference between any two consecutive terms is constant.
• Additive pattern: Each term is obtained by adding a fixed constant to the previous term.
• Linear relationship: The terms of an arithmetic sequence are related by a linear equation.

Q: How do I determine if a sequence is arithmetic?

A: To determine if a sequence is arithmetic, you need to check if the difference between consecutive terms is constant. You can do this by subtracting each term from the previous term and checking if the result is the same.

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:

$$a_n = a_1 + (n - 1)d$$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

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

A: To find the common difference of an arithmetic sequence, you need to subtract each term from the previous term and check if the result is the same.

Q: What are some examples of arithmetic sequences?

A: Some examples of arithmetic sequences include:

• $$2, 5, 8, 11, \ldots$$

• $$-3, 4, 11, 18, \ldots$$

• $$10, 15, 20, 25, \ldots$$

Q: What are some real-world applications of arithmetic sequences?

A: Some real-world applications of arithmetic sequences include:

• Finance: Arithmetic sequences are used to calculate interest rates, investments, and annuities.
• Science: Arithmetic sequences are used to model population growth, chemical reactions, and physical phenomena.
• Engineering: Arithmetic sequences are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: How do I use arithmetic sequences in real-world problems?

A: To use arithmetic sequences in real-world problems, you need to identify the pattern of the sequence and use it to solve the problem. For example, if you are calculating the future value of an investment, you can use an arithmetic sequence to model the growth of the investment.

Q: What are some common mistakes to avoid when working with arithmetic sequences?

A: Some common mistakes to avoid when working with arithmetic sequences include:

• Not checking if the sequence is arithmetic: Make sure to check if the sequence is arithmetic before using the formula for the nth term.
• Not using the correct formula: Make sure to use the correct formula for the nth term, which is $a_n = a_1 + (n - 1)d$.
• Not checking for errors: Make sure to check for errors in your calculations and double-check your answers.

Q: How do I determine if an arithmetic sequence is finite or infinite?

A: To determine if an arithmetic sequence is finite or infinite, you need to check if the sequence has a last term. If the sequence has a last term, it is finite. If the sequence does not have a last term, it is infinite.

Q: What are some advanced topics related to arithmetic sequences?

A: Some advanced topics related to arithmetic sequences include:

• Arithmetic series: An arithmetic series is the sum of the terms of an arithmetic sequence.
• Arithmetic progression: An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant.
• Arithmetic mean: The arithmetic mean is the average of a set of numbers, which can be calculated using an arithmetic sequence.

Conclusion

In this article, we answered some frequently asked questions about arithmetic sequences, including their properties, applications, and examples. We also discussed some common mistakes to avoid when working with arithmetic sequences and some advanced topics related to arithmetic sequences.