Which Of The Following Is An Arithmetic Sequence Brainly

An arithmetic sequence, at its core, is a progression of numbers where the difference between any two consecutive terms remains constant. This constant difference is known as the common difference, and it's the defining characteristic that sets arithmetic sequences apart from other types of sequences. Identifying whether a given sequence is arithmetic requires a careful examination of the differences between its terms.

Defining Arithmetic Sequences

To truly understand how to identify an arithmetic sequence, let's delve deeper into its definition and properties.

• Terms: Each number in the sequence is called a term. We often denote the first term as a₁, the second term as a₂, and so on.

• Common Difference (d): The common difference is the constant value added to each term to obtain the next term. Mathematically, d = a₂ - a₁ = a₃ - a₂ = a₄ - a₃ and so on.

• General Formula: The nth term of an arithmetic sequence can be calculated using the following formula:

  • aₙ = a₁ + (n - 1)d

  This formula allows you to find any term in the sequence if you know the first term (a₁) and the common difference (d).

Identifying Arithmetic Sequences: A Step-by-Step Guide

The most reliable way to determine if a sequence is arithmetic is to check if there is a constant difference between consecutive terms. Here's a detailed step-by-step guide:

1. Calculate the Differences: Subtract each term from its subsequent term. In other words, calculate:

   • a₂ - a₁
   • a₃ - a₂
   • a₄ - a₃
   • And so on, for as many terms as are provided in the sequence.

2. Compare the Differences: Examine the differences you calculated in the previous step. Are they all the same?

3. Conclusion:

   • If the differences are the same (constant): The sequence is arithmetic. The constant difference you found is the common difference (d) of the sequence.
   • If the differences are not the same: The sequence is not arithmetic.

Examples: Putting the Steps into Practice

Let's illustrate this process with several examples. We'll analyze different sequences and determine whether they are arithmetic or not.

Example 1:

Sequence: 2, 5, 8, 11, 14,...

1. Calculate the Differences:

   • 5 - 2 = 3
   • 8 - 5 = 3
   • 11 - 8 = 3
   • 14 - 11 = 3

2. Compare the Differences: All the differences are equal to 3.

3. Conclusion: This sequence is arithmetic. The common difference, d, is 3.

Example 2:

Sequence: 1, 4, 9, 16, 25,...

1. Calculate the Differences:

   • 4 - 1 = 3
   • 9 - 4 = 5
   • 16 - 9 = 7
   • 25 - 16 = 9

2. Compare the Differences: The differences are 3, 5, 7, and 9. They are not the same.

3. Conclusion: This sequence is not arithmetic. (This is actually a sequence of square numbers).

Example 3:

Sequence: 10, 7, 4, 1, -2,...

1. Calculate the Differences:

   • 7 - 10 = -3
   • 4 - 7 = -3
   • 1 - 4 = -3
   • -2 - 1 = -3

2. Compare the Differences: All the differences are equal to -3.

3. Conclusion: This sequence is arithmetic. The common difference, d, is -3. Notice that the common difference can be negative, resulting in a decreasing sequence.

Example 4:

Sequence: 2, 4, 8, 16, 32,...

1. Calculate the Differences:

   • 4 - 2 = 2
   • 8 - 4 = 4
   • 16 - 8 = 8
   • 32 - 16 = 16

2. Compare the Differences: The differences are 2, 4, 8, and 16. They are not the same.

3. Conclusion: This sequence is not arithmetic. (This is a geometric sequence where each term is multiplied by 2).

Example 5:

Sequence: 1/2, 1, 3/2, 2, 5/2,...

1. Calculate the Differences:

   • 1 - 1/2 = 1/2
   • 3/2 - 1 = 1/2
   • 2 - 3/2 = 1/2
   • 5/2 - 2 = 1/2

2. Compare the Differences: All the differences are equal to 1/2.

3. Conclusion: This sequence is arithmetic. The common difference, d, is 1/2.

Common Mistakes to Avoid

When identifying arithmetic sequences, be aware of these common pitfalls:

• Assuming a pattern after only a few terms: Just because the first few terms seem to follow a pattern doesn't guarantee that the entire sequence is arithmetic. You must check the differences between several consecutive terms.
• Confusing arithmetic and geometric sequences: Arithmetic sequences have a constant difference between terms, while geometric sequences have a constant ratio between terms. Make sure you're checking the correct relationship.
• Ignoring negative numbers and fractions: Arithmetic sequences can contain negative numbers, fractions, and decimals. Don't let these distract you from applying the same principles.
• Calculating differences in the wrong order: Always subtract a term from the subsequent term (e.g., a₂ - a₁, not a₁ - a₂). Reversing the order will give you the negative of the common difference, which can lead to confusion.

Beyond Simple Sequences: Arithmetic Progressions in Real-World Applications

Arithmetic sequences aren't just abstract mathematical concepts; they appear in various real-world scenarios. Understanding them can help you model and analyze these situations.

• Simple Interest: If you deposit money in a bank account with simple interest, the amount of money you have each year forms an arithmetic sequence. The initial deposit is a₁, and the annual interest is the common difference, d.

• Depreciation: The value of an asset that depreciates linearly (by a fixed amount each year) forms an arithmetic sequence.

• Staircases: The height of each step in a staircase, assuming all steps are the same height, forms an arithmetic sequence.

• Patterns in Nature: While not always perfectly arithmetic, some patterns in nature, like the arrangement of leaves on a stem in certain plants, can approximate arithmetic progressions.

• ** равномерно accelerated Motion:** In physics, the distance traveled by an object undergoing uniformly accelerated motion over equal time intervals forms an arithmetic sequence.

Advanced Applications and Related Concepts

While the core concept of arithmetic sequences is straightforward, they connect to more advanced mathematical ideas:

• Arithmetic Series: An arithmetic series is the sum of the terms in an arithmetic sequence. The sum of the first n terms of an arithmetic series is given by:

  • Sₙ = n/2 [2a₁ + (n - 1)d] or Sₙ = n/2 (a₁ + aₙ)

• Linear Functions: There's a strong relationship between arithmetic sequences and linear functions. The graph of an arithmetic sequence is a set of discrete points that lie on a straight line. The common difference corresponds to the slope of the line.

• Difference Equations: Arithmetic sequences can be described by a simple type of difference equation, which relates a term to its previous term(s).

• Mathematical Induction: Proofs involving arithmetic sequences often utilize mathematical induction, a powerful technique for proving statements that hold for all natural numbers.

Practice Problems: Test Your Understanding

To solidify your understanding of arithmetic sequences, try working through these practice problems:

1. Determine if the following sequence is arithmetic: 3, 7, 11, 15, 19,... If so, find the common difference and the 10th term.

2. Determine if the following sequence is arithmetic: 1, 2, 4, 8, 16,...

3. The first term of an arithmetic sequence is 5, and the common difference is -2. Find the first five terms of the sequence.

4. Find the common difference of an arithmetic sequence if the 3rd term is 8 and the 7th term is 20.

5. The 5th term of an arithmetic sequence is 12, and the 12th term is 33. Find the first term and the common difference.

Answers to Practice Problems

1. Arithmetic: Yes. d = 4. a₁₀ = 39.
2. Arithmetic: No.
3. First five terms: 5, 3, 1, -1, -3.
4. Common difference: d = 3.
5. First term: a₁ = 0. Common difference: d = 3.

Conclusion

Identifying whether a sequence is arithmetic is a fundamental skill in mathematics. By understanding the definition of an arithmetic sequence, following the step-by-step guide for checking the common difference, and avoiding common mistakes, you can confidently determine if a given sequence belongs to this important class of mathematical progressions. Furthermore, recognizing the real-world applications and connections to more advanced concepts enhances the value of this knowledge and empowers you to analyze and solve a wider range of problems. So, practice diligently, explore different examples, and embrace the power of arithmetic sequences! Remember to always check for a constant difference!