Match The Parts Of The Following Function With Their Correct Definitions.${ A(n) = A + (n-1)d }$1. { D $}$ - Common Difference2. { N $}$ - Number Of The { N $}$th Term3. { A $}$ - Value

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In mathematics, an arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. The arithmetic sequence function is a mathematical representation of this sequence, and it is used to find the nth term of the sequence. In this article, we will match the parts of the arithmetic sequence function with their correct definitions.

The Arithmetic Sequence Function

The arithmetic sequence function is given by the formula:

A(n)=a+(n−1)d{ A(n) = a + (n-1)d }

Where:

  • A(n) is the nth term of the arithmetic sequence
  • a is the first term of the arithmetic sequence
  • d is the common difference between consecutive terms
  • n is the term number

Matching the Parts of the Function

Now, let's match the parts of the function with their correct definitions.

1. d - Common Difference

The common difference is the constant difference between any two consecutive terms in an arithmetic sequence. It is denoted by the symbol d in the arithmetic sequence function. The common difference is the key characteristic that distinguishes an arithmetic sequence from other types of sequences.

For example, consider an arithmetic sequence with the first term a = 2 and the common difference d = 3. The sequence would be: 2, 5, 8, 11, 14, ...

In this example, the common difference d is 3, which means that each term is obtained by adding 3 to the previous term.

2. n - Number of the nth term

The term number n represents the position of the term in the arithmetic sequence. It is used to find the nth term of the sequence using the arithmetic sequence function.

For example, if we want to find the 5th term of the arithmetic sequence with the first term a = 2 and the common difference d = 3, we would use the term number n = 5 in the arithmetic sequence function.

3. a - Value of the first term

The first term a is the initial term of the arithmetic sequence. It is the starting point of the sequence, and all other terms are obtained by adding the common difference to the previous term.

For example, in the arithmetic sequence 2, 5, 8, 11, 14, ..., the first term a is 2.

Example Problems

Now that we have matched the parts of the function with their correct definitions, let's solve some example problems to illustrate how to use the arithmetic sequence function.

Example 1

Find the 10th term of the arithmetic sequence with the first term a = 5 and the common difference d = 2.

Using the arithmetic sequence function, we get:

A(10)=5+(10−1)2{ A(10) = 5 + (10-1)2 } A(10)=5+9(2){ A(10) = 5 + 9(2) } A(10)=5+18{ A(10) = 5 + 18 } A(10)=23{ A(10) = 23 }

Therefore, the 10th term of the arithmetic sequence is 23.

Example 2

Find the value of the first term a in the arithmetic sequence with the 5th term A(5) = 17 and the common difference d = 4.

Using the arithmetic sequence function, we get:

17=a+(5−1)4{ 17 = a + (5-1)4 } 17=a+4(4){ 17 = a + 4(4) } 17=a+16{ 17 = a + 16 } a=17−16{ a = 17 - 16 } a=1{ a = 1 }

Therefore, the value of the first term a is 1.

Conclusion

In this article, we have matched the parts of the arithmetic sequence function with their correct definitions. We have also solved some example problems to illustrate how to use the arithmetic sequence function to find the nth term of an arithmetic sequence. The arithmetic sequence function is a powerful tool in mathematics, and it has numerous applications in various fields, including science, engineering, and economics.

Frequently Asked Questions

Q: What is the arithmetic sequence function?

A: The arithmetic sequence function is a mathematical representation of an arithmetic sequence, which is a sequence of numbers in which the difference between any two consecutive terms is constant.

Q: What is the common difference?

A: The common difference is the constant difference between any two consecutive terms in an arithmetic sequence.

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

A: To find the nth term of an arithmetic sequence, you can use the arithmetic sequence function, which is given by the formula: A(n) = a + (n-1)d.

Q: What is the first term of an arithmetic sequence?

A: The first term of an arithmetic sequence is the initial term, which is denoted by the symbol a in the arithmetic sequence function.

Glossary

  • Arithmetic sequence: A sequence of numbers in which the difference between any two consecutive terms is constant.
  • Common difference: The constant difference between any two consecutive terms in an arithmetic sequence.
  • Term number: The position of the term in the arithmetic sequence.
  • First term: The initial term of the arithmetic sequence.
  • Arithmetic sequence function: A mathematical representation of an arithmetic sequence, which is given by the formula: A(n) = a + (n-1)d.
    Arithmetic Sequence Function Q&A =====================================

In this article, we will continue to explore the arithmetic sequence function and answer some frequently asked questions.

Q: What is the formula for the arithmetic sequence function?

A: The formula for the arithmetic sequence function is:

A(n)=a+(n−1)d{ A(n) = a + (n-1)d }

Where:

  • A(n) is the nth term of the arithmetic sequence
  • a is the first term of the arithmetic sequence
  • d is the common difference between consecutive terms
  • n is the term number

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

A: To find the nth term of an arithmetic sequence, you can use the arithmetic sequence function. Simply plug in the values of a, d, and n into the formula:

A(n)=a+(n−1)d{ A(n) = a + (n-1)d }

For example, if you want to find the 5th term of an arithmetic sequence with the first term a = 2 and the common difference d = 3, you would use the formula:

A(5)=2+(5−1)3{ A(5) = 2 + (5-1)3 } A(5)=2+4(3){ A(5) = 2 + 4(3) } A(5)=2+12{ A(5) = 2 + 12 } A(5)=14{ A(5) = 14 }

Therefore, the 5th term of the arithmetic sequence is 14.

Q: What is the significance of the common difference?

A: The common difference is the key characteristic that distinguishes an arithmetic sequence from other types of sequences. It is the constant difference between any two consecutive terms in the sequence.

For example, in the arithmetic sequence 2, 5, 8, 11, 14, ..., the common difference is 3. This means that each term is obtained by adding 3 to the previous term.

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){ d = A(n) - A(n-1) }

Where:

  • d is the common difference
  • A(n) is the nth term of the arithmetic sequence
  • A(n-1) is the (n-1)th term of the arithmetic sequence

For example, if you want to find the common difference of an arithmetic sequence with the 5th term A(5) = 14 and the 4th term A(4) = 11, you would use the formula:

d=14−11{ d = 14 - 11 } d=3{ d = 3 }

Therefore, the common difference of the arithmetic sequence is 3.

Q: What is the significance of the term number?

A: The term number is the position of the term in the arithmetic sequence. It is used to find the nth term of the sequence using the arithmetic sequence function.

For example, if you want to find the 5th term of an arithmetic sequence, you would use the term number n = 5 in the arithmetic sequence function.

Q: How do I find the term number of an arithmetic sequence?

A: To find the term number of an arithmetic sequence, you can use the formula:

n=A(n)−ad+1{ n = \frac{A(n) - a}{d} + 1 }

Where:

  • n is the term number
  • A(n) is the nth term of the arithmetic sequence
  • a is the first term of the arithmetic sequence
  • d is the common difference

For example, if you want to find the term number of an arithmetic sequence with the 5th term A(5) = 14, the first term a = 2, and the common difference d = 3, you would use the formula:

n=14−23+1{ n = \frac{14 - 2}{3} + 1 } n=123+1{ n = \frac{12}{3} + 1 } n=4+1{ n = 4 + 1 } n=5{ n = 5 }

Therefore, the term number of the arithmetic sequence is 5.

Q: What is the relationship between the arithmetic sequence function and the geometric sequence function?

A: The arithmetic sequence function and the geometric sequence function are two different types of sequence functions. The arithmetic sequence function is used to find the nth term of an arithmetic sequence, while the geometric sequence function is used to find the nth term of a geometric sequence.

The arithmetic sequence function is given by the formula:

A(n)=a+(n−1)d{ A(n) = a + (n-1)d }

Where:

  • A(n) is the nth term of the arithmetic sequence
  • a is the first term of the arithmetic sequence
  • d is the common difference between consecutive terms
  • n is the term number

The geometric sequence function is given by the formula:

G(n)=a⋅rn−1{ G(n) = a \cdot r^{n-1} }

Where:

  • G(n) is the nth term of the geometric sequence
  • a is the first term of the geometric sequence
  • r is the common ratio between consecutive terms
  • n is the term number

Q: What are some real-world applications of the arithmetic sequence function?

A: The arithmetic sequence function has numerous real-world applications, including:

  • Finance: The arithmetic sequence function is used to calculate interest rates and investment returns.
  • Science: The arithmetic sequence function is used to model population growth and decay.
  • Engineering: The arithmetic sequence function is used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: The arithmetic sequence function is used to model economic growth and inflation.

Conclusion

In this article, we have answered some frequently asked questions about the arithmetic sequence function. We have also explored the significance of the common difference, the term number, and the relationship between the arithmetic sequence function and the geometric sequence function. The arithmetic sequence function is a powerful tool in mathematics, and it has numerous real-world applications.