Given The Functions F ( N ) = 25 F(n) = 25 F ( N ) = 25 And G ( N ) = 3 ( N − 1 G(n) = 3(n-1 G ( N ) = 3 ( N − 1 ], Combine Them To Create An Arithmetic Sequence A N A_n A N ​ And Solve For The 12th Term.A. A N = 25 − 3 ( N − 1 ) ; A 12 = − 11 A_n = 25 - 3(n-1) ; A_{12} = -11 A N ​ = 25 − 3 ( N − 1 ) ; A 12 ​ = − 11 B. $a_n = 25 - 3(n-1) ; A_{12} =

Introduction

In mathematics, functions are used to describe relationships between variables. When we combine two functions, we can create new sequences and patterns. In this article, we will explore how to combine two given functions to create an arithmetic sequence and solve for the 12th term.

Understanding the Functions

The two given functions are:

• $f(n) = 25$
• $g(n) = 3(n-1)$

These functions take a single input, $n$, and produce a corresponding output. The function $f(n)$ always returns the value 25, regardless of the input $n$. The function $g(n)$, on the other hand, returns a value that depends on the input $n$. Specifically, it returns 3 times the value of $n-1$.

Combining the Functions

To create an arithmetic sequence, we need to combine the two functions in a way that produces a sequence of numbers. We can do this by adding or subtracting the two functions. Let's try subtracting the function $g(n)$ from the function $f(n)$:

$a_n = f(n) - g(n)$

Substituting the given functions, we get:

$a_n = 25 - 3(n-1)$

This is the combined function that produces the arithmetic sequence.

Solving for the 12th Term

Now that we have the combined function, we can solve for the 12th term of the sequence. To do this, we need to plug in $n=12$ into the combined function:

$a_{12} = 25 - 3(12-1)$

Simplifying the expression, we get:

$a_{12} = 25 - 3(11)$

$a_{12} = 25 - 33$

$a_{12} = -8$

Therefore, the 12th term of the sequence is $-8$.

Conclusion

In this article, we combined two given functions to create an arithmetic sequence and solved for the 12th term. We started with the functions $f(n) = 25$ and $g(n) = 3(n-1)$ and combined them by subtracting $g(n)$ from $f(n)$. The resulting combined function was $a_n = 25 - 3(n-1)$. We then solved for the 12th term by plugging in $n=12$ into the combined function and simplifying the expression. The result was $a_{12} = -8$.

Arithmetic Sequences and Series

Arithmetic sequences and series are fundamental concepts in mathematics. An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term. An arithmetic series is the sum of an arithmetic sequence.

Arithmetic sequences and series have many real-world applications, including finance, physics, and engineering. They are used to model population growth, financial investments, and physical phenomena such as the motion of objects.

Properties of Arithmetic Sequences

Arithmetic sequences have several important properties, including:

• Common difference: The common difference is the fixed constant that is added to each term to obtain the next term.
• First term: The first term is the first term of the sequence.
• Last term: The last term is the last term of the sequence.
• Number of terms: The number of terms is the total number of terms in the sequence.

Solving for the nth Term of an Arithmetic Sequence

To solve for the nth term of an arithmetic sequence, we can use the formula:

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

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

Example: Solving for the 12th Term of an Arithmetic Sequence

Suppose we have an arithmetic sequence with a first term of 2 and a common difference of 3. We want to find the 12th term of the sequence. Using the formula, we get:

$a_{12} = 2 + (12-1)3$

$a_{12} = 2 + 33$

$a_{12} = 35$

Therefore, the 12th term of the sequence is 35.

Conclusion

Introduction

Arithmetic sequences and series are fundamental concepts in mathematics that have numerous real-world applications. In this article, we will provide a Q&A guide to help you understand and work with arithmetic sequences and series.

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.

Q: What is an arithmetic series?

A: An arithmetic series is the sum of an arithmetic sequence.

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 can use the formula:

$d = a_n - a_{n-1}$

where $a_n$ is the nth term and $a_{n-1}$ is the (n-1)th term.

Q: How do I find the sum of an arithmetic series?

A: To find the sum of an arithmetic series, you can use the formula:

$S_n = \frac{n}{2}(a_1 + a_n)$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $a_n$ is the nth term, and $n$ is the number of terms.

Q: What is the formula for the sum of an infinite arithmetic series?

A: The formula for the sum of an infinite arithmetic series is:

$S = \frac{a_1}{1 - r}$

where $S$ is the sum of the infinite series, $a_1$ is the first term, and $r$ is the common ratio.

Q: How do I determine if an arithmetic sequence is increasing or decreasing?

A: To determine if an arithmetic sequence is increasing or decreasing, you can look at the common difference. If the common difference is positive, the sequence is increasing. If the common difference is negative, the sequence is decreasing.

Q: What is the relationship between arithmetic sequences and geometric sequences?

A: Arithmetic sequences and geometric sequences are both types of sequences, but they have different properties. Arithmetic sequences have a common difference, while geometric sequences have a common ratio.

Q: How do I use arithmetic sequences and series in real-world applications?

A: Arithmetic sequences and series have numerous real-world applications, including finance, physics, and engineering. They are used to model population growth, financial investments, and physical phenomena such as the motion of objects.

Conclusion

In this article, we provided a Q&A guide to help you understand and work with arithmetic sequences and series. We covered topics such as the formula for the nth term, the common difference, the sum of an arithmetic series, and the relationship between arithmetic sequences and geometric sequences. We also discussed real-world applications of arithmetic sequences and series.

Frequently Asked Questions

• Q: What is the difference between an arithmetic sequence and a geometric sequence? A: An arithmetic sequence has a common difference, while a geometric sequence has a common ratio.
• Q: How do I find the sum of an infinite arithmetic series? A: You can use the formula $S = \frac{a_1}{1 - r}$, where $S$ is the sum of the infinite series, $a_1$ is the first term, and $r$ is the common ratio.
• Q: What is the relationship between arithmetic sequences and financial investments? A: Arithmetic sequences are used to model financial investments, such as compound interest and annuities.
• Q: How do I use arithmetic sequences and series in physics and engineering? A: Arithmetic sequences and series are used to model physical phenomena such as the motion of objects and population growth.

Glossary

• Arithmetic sequence: A sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.
• Arithmetic series: The sum of an arithmetic sequence.
• Common difference: The fixed constant that is added to each term to obtain the next term.
• Common ratio: The fixed constant that is multiplied by each term to obtain the next term.
• Infinite arithmetic series: An arithmetic series with an infinite number of terms.
• Sum of an arithmetic series: The sum of the first n terms of an arithmetic sequence.