Select The Correct Answer.Which Explicit Function Defines This Arithmetic Sequence? $-351, -343, -335, -327, -319$A. $f(n) = 8n - 351$ B. $f(n) = -8n - 351$ C. $f(n) = -8n + 359$ D. $f(n) = 8n - 359$

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. Arithmetic sequences are an essential concept in mathematics, and they have numerous applications in various fields, including finance, engineering, and computer science.

Identifying the Common Difference

To determine the explicit function that defines an arithmetic sequence, we need to identify the common difference. The common difference can be found by subtracting any term from its preceding term. In the given sequence, $-351, -343, -335, -327, -319$, we can find the common difference by subtracting the first term from the second term:

$$-343 - (-351) = -343 + 351 = 8$$

The common difference is $8$. This means that each term in the sequence is obtained by adding $8$ to the previous term.

Determining the Explicit Function

Now that we have identified the common difference, we can determine the explicit function that defines the arithmetic sequence. The explicit function is a formula that takes an integer $n$ as input and returns the $n$th term of the sequence. The general form of an explicit function for an arithmetic sequence is:

$$f(n) = an + b$$

where $a$ is the common difference and $b$ is the first term of the sequence.

Evaluating the Options

Let's evaluate the options to determine which one defines the given arithmetic sequence.

Option A: $f(n) = 8n - 351$

To determine if this option is correct, we need to substitute the first term of the sequence, $-351$, into the formula and check if it matches the first term of the sequence.

$$f(1) = 8(1) - 351 = 8 - 351 = -343$$

This matches the second term of the sequence. However, we need to check if the formula produces the correct sequence.

Let's substitute $n = 2$ into the formula:

$$f(2) = 8(2) - 351 = 16 - 351 = -335$$

This matches the third term of the sequence. Let's continue this process to see if the formula produces the correct sequence.

$$f(3) = 8(3) - 351 = 24 - 351 = -327$$

$$f(4) = 8(4) - 351 = 32 - 351 = -319$$

This matches the fourth and fifth terms of the sequence. Therefore, option A is correct.

Option B: $f(n) = -8n - 351$

To determine if this option is correct, we need to substitute the first term of the sequence, $-351$, into the formula and check if it matches the first term of the sequence.

$$f(1) = -8(1) - 351 = -8 - 351 = -359$$

This does not match the first term of the sequence. Therefore, option B is incorrect.

Option C: $f(n) = -8n + 359$

To determine if this option is correct, we need to substitute the first term of the sequence, $-351$, into the formula and check if it matches the first term of the sequence.

$$f(1) = -8(1) + 359 = -8 + 359 = 351$$

This does not match the first term of the sequence. Therefore, option C is incorrect.

Option D: $f(n) = 8n - 359$

To determine if this option is correct, we need to substitute the first term of the sequence, $-351$, into the formula and check if it matches the first term of the sequence.

$$f(1) = 8(1) - 359 = 8 - 359 = -351$$

This matches the first term of the sequence. However, we need to check if the formula produces the correct sequence.

Let's substitute $n = 2$ into the formula:

$$f(2) = 8(2) - 359 = 16 - 359 = -343$$

This matches the second term of the sequence. Let's continue this process to see if the formula produces the correct sequence.

$$f(3) = 8(3) - 359 = 24 - 359 = -335$$

$$f(4) = 8(4) - 359 = 32 - 359 = -327$$

$$f(5) = 8(5) - 359 = 40 - 359 = -319$$

This matches the third, fourth, and fifth terms of the sequence. Therefore, option D is correct.

Conclusion

$$$$$$$$$$

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. This constant difference is called the common difference.

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

A: To find the common difference, subtract any term from its preceding term. For example, if the sequence is $-351, -343, -335, -327, -319$, you can find the common difference by subtracting the first term from the second term:

$$-343 - (-351) = -343 + 351 = 8$$

Q: What is the explicit function of an arithmetic sequence?

A: The explicit function is a formula that takes an integer $n$ as input and returns the $n$th term of the sequence. The general form of an explicit function for an arithmetic sequence is:

$$f(n) = an + b$$

where $a$ is the common difference and $b$ is the first term of the sequence.

Q: How do I determine the explicit function of an arithmetic sequence?

A: To determine the explicit function, you need to identify the common difference and the first term of the sequence. Then, use the formula:

$$f(n) = an + b$$

where $a$ is the common difference and $b$ is the first term of the sequence.

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

A: The common difference is the constant difference between any two consecutive terms in an arithmetic sequence. It is essential in determining the explicit function of the sequence.

Q: Can an arithmetic sequence have a negative common difference?

A: Yes, an arithmetic sequence can have a negative common difference. For example, the sequence $-351, -343, -335, -327, -319$ has a common difference of $8$, which is positive. However, the sequence $-351, -343, -335, -327, -319$ has a common difference of $-8$, which is negative.

Q: Can an arithmetic sequence have a zero common difference?

A: No, an arithmetic sequence cannot have a zero common difference. If the common difference is zero, the sequence would be a constant sequence, not an arithmetic sequence.

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

A: To determine if a sequence is an arithmetic sequence, check if the difference between any two consecutive terms is constant. If the difference is constant, the sequence is an arithmetic sequence.

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

A: Arithmetic sequences have numerous real-world applications, including:

• Finance: Compound interest and annuities
• Engineering: Motion and vibration analysis
• Computer Science: Algorithm design and analysis
• Biology: Population growth and decline

Q: Can I use arithmetic sequences to model real-world phenomena?

A: Yes, arithmetic sequences can be used to model real-world phenomena, such as population growth, compound interest, and motion. However, the sequence must be an arithmetic sequence, and the common difference must be constant.

Q: How do I use arithmetic sequences to solve problems?

A: To use arithmetic sequences to solve problems, identify the common difference and the first term of the sequence. Then, use the explicit function to find the $n$th term of the sequence. Finally, use the $n$th term to solve the problem.

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:

• Assuming a sequence is an arithmetic sequence when it is not
• Using the wrong common difference
• Not checking if the sequence is an arithmetic sequence before using it
• Not using the explicit function to find the $n$th term of the sequence

Conclusion

In conclusion, arithmetic sequences and explicit functions are essential concepts in mathematics. Understanding arithmetic sequences and explicit functions can help you solve problems in finance, engineering, computer science, and biology. By following the steps outlined in this article, you can determine the explicit function of an arithmetic sequence and use it to solve problems.