QUESTION 22.1 Determine Which Term Of The Sequence ${ 23, 21, 19, \ldots\$} Is { -47$}$. (3 Points)2.2 In The Quadratic Sequence ${ 4, X, Y, -11, \ldots\$} , The First Three Terms Of The First Differences Are [$2p - 4,

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Introduction

Sequences are a fundamental concept in mathematics, and they play a crucial role in various mathematical disciplines, including algebra, geometry, and calculus. A sequence is a list of numbers in a specific order, and it can be defined by a formula or rule. Quadratic sequences, in particular, are sequences where each term is obtained by adding or subtracting a constant value to the previous term, and the difference between consecutive terms is a quadratic expression. In this article, we will explore quadratic sequences, their properties, and how to determine the term of a quadratic sequence.

What is a Quadratic Sequence?

A quadratic sequence is a sequence where each term is obtained by adding or subtracting a constant value to the previous term, and the difference between consecutive terms is a quadratic expression. The general form of a quadratic sequence is:

a, a + d, a + 2d, a + 3d, ...

where a is the first term, and d is the common difference.

Properties of Quadratic Sequences

Quadratic sequences have several properties that make them useful in mathematics. Some of the key properties of quadratic sequences include:

  • Constant difference: The difference between consecutive terms is constant.
  • Quadratic expression: The difference between consecutive terms is a quadratic expression.
  • Symmetry: Quadratic sequences are symmetric about the midpoint of the sequence.

Determining the Term of a Quadratic Sequence

To determine the term of a quadratic sequence, we need to know the first term and the common difference. The formula for the nth term of a quadratic sequence is:

an = a + (n - 1)d

where an is the nth term, a is the first term, n is the term number, and d is the common difference.

Example 1: Determining the Term of a Quadratic Sequence

Suppose we have a quadratic sequence with the first term a = 23 and the common difference d = -2. We want to find the term that is equal to -47.

Using the formula for the nth term, we get:

an = 23 + (n - 1)(-2)

We want to find the value of n such that an = -47. We can set up an equation and solve for n:

23 + (n - 1)(-2) = -47

Simplifying the equation, we get:

-2n + 25 = -47

Subtracting 25 from both sides, we get:

-2n = -72

Dividing both sides by -2, we get:

n = 36

Therefore, the term that is equal to -47 is the 36th term.

Example 2: Determining the Term of a Quadratic Sequence

Suppose we have a quadratic sequence with the first term a = 4 and the common difference d = 2p - 4. We want to find the term that is equal to -11.

Using the formula for the nth term, we get:

an = 4 + (n - 1)(2p - 4)

We want to find the value of n such that an = -11. We can set up an equation and solve for n:

4 + (n - 1)(2p - 4) = -11

Simplifying the equation, we get:

2np - 4n + 4p - 8 = -15

Rearranging the terms, we get:

2np - 4n + 4p = -7

Factoring out the common term, we get:

(2p - 4)n + 4p = -7

Subtracting 4p from both sides, we get:

(2p - 4)n = -4p - 7

Dividing both sides by (2p - 4), we get:

n = (-4p - 7) / (2p - 4)

Simplifying the expression, we get:

n = (-2p - 7/2) / (p - 2)

Therefore, the term that is equal to -11 is the nth term, where n is given by the expression above.

Conclusion

Quadratic sequences are a fundamental concept in mathematics, and they play a crucial role in various mathematical disciplines. In this article, we explored quadratic sequences, their properties, and how to determine the term of a quadratic sequence. We also provided two examples of determining the term of a quadratic sequence. By understanding and solving quadratic sequences, we can gain a deeper understanding of mathematical concepts and develop problem-solving skills.

References

Further Reading

Frequently Asked Questions

Q: What is a quadratic sequence?

A: A quadratic sequence is a sequence where each term is obtained by adding or subtracting a constant value to the previous term, and the difference between consecutive terms is a quadratic expression.

Q: What are the properties of quadratic sequences?

A: Quadratic sequences have several properties that make them useful in mathematics. Some of the key properties of quadratic sequences include:

  • Constant difference: The difference between consecutive terms is constant.
  • Quadratic expression: The difference between consecutive terms is a quadratic expression.
  • Symmetry: Quadratic sequences are symmetric about the midpoint of the sequence.

Q: How do I determine the term of a quadratic sequence?

A: To determine the term of a quadratic sequence, you need to know the first term and the common difference. The formula for the nth term of a quadratic sequence is:

an = a + (n - 1)d

where an is the nth term, a is the first term, n is the term number, and d is the common difference.

Q: What is the formula for the nth term of a quadratic sequence?

A: The formula for the nth term of a quadratic sequence is:

an = a + (n - 1)d

where an is the nth term, a is the first term, n is the term number, and d is the common difference.

Q: How do I find the value of n in the formula for the nth term of a quadratic sequence?

A: To find the value of n in the formula for the nth term of a quadratic sequence, you need to set up an equation and solve for n. The equation is:

an = a + (n - 1)d

You can solve for n by rearranging the equation and isolating n.

Q: What is the difference between a quadratic sequence and a linear sequence?

A: A quadratic sequence is a sequence where each term is obtained by adding or subtracting a constant value to the previous term, and the difference between consecutive terms is a quadratic expression. A linear sequence, on the other hand, is a sequence where each term is obtained by adding or subtracting a constant value to the previous term, and the difference between consecutive terms is a linear expression.

Q: Can you provide an example of a quadratic sequence?

A: Yes, here is an example of a quadratic sequence:

2, 5, 8, 11, 14, ...

This is a quadratic sequence because each term is obtained by adding 3 to the previous term, and the difference between consecutive terms is a quadratic expression.

Q: Can you provide an example of a linear sequence?

A: Yes, here is an example of a linear sequence:

2, 4, 6, 8, 10, ...

This is a linear sequence because each term is obtained by adding 2 to the previous term, and the difference between consecutive terms is a linear expression.

Q: How do I know if a sequence is quadratic or linear?

A: To determine if a sequence is quadratic or linear, you need to examine the difference between consecutive terms. If the difference between consecutive terms is a quadratic expression, then the sequence is quadratic. If the difference between consecutive terms is a linear expression, then the sequence is linear.

Q: Can you provide a formula for the sum of a quadratic sequence?

A: Yes, the formula for the sum of a quadratic sequence is:

S = (n/2)(2a + (n - 1)d)

where S is the sum, n is the number of terms, a is the first term, and d is the common difference.

Q: Can you provide a formula for the product of a quadratic sequence?

A: Yes, the formula for the product of a quadratic sequence is:

P = a^n

where P is the product, n is the number of terms, and a is the first term.

Conclusion

Quadratic sequences are a fundamental concept in mathematics, and they play a crucial role in various mathematical disciplines. In this article, we explored quadratic sequences, their properties, and how to determine the term of a quadratic sequence. We also provided several examples and formulas to help you understand and work with quadratic sequences. By understanding and solving quadratic sequences, you can gain a deeper understanding of mathematical concepts and develop problem-solving skills.

References

Further Reading