Finding The Formula For An Arithmetic Sequence 25, 31, 37, 43, 49

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Let's explore the fascinating world of arithmetic sequences and discover how to represent them with a formula. In this article, we will analyze the given arithmetic sequence, 25, 31, 37, 43, 49..., and determine the correct formula that defines it. We will delve into the fundamental concepts of arithmetic sequences, including the first term, the common difference, and the general formula, to provide a comprehensive understanding of the topic. We will examine each option provided, meticulously dissecting its structure and comparing it to the observed pattern in the sequence. By the end of this exploration, you will not only be able to identify the correct formula but also possess a solid grasp of how to construct formulas for any arithmetic sequence.

Understanding Arithmetic Sequences

To begin, let's define what an arithmetic sequence truly is. An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'. In simpler terms, you get the next number in the sequence by adding the same value to the previous number. For example, in the sequence 2, 4, 6, 8, 10, the common difference is 2 because each term is obtained by adding 2 to the previous term. Identifying the common difference is crucial in understanding and representing arithmetic sequences.

Now, let's consider the given sequence: 25, 31, 37, 43, 49.... To determine if it's an arithmetic sequence, we need to check if the difference between consecutive terms is constant. The difference between 31 and 25 is 6. The difference between 37 and 31 is also 6. Similarly, 43 - 37 = 6 and 49 - 43 = 6. Since the difference between consecutive terms is consistently 6, we can confirm that this is indeed an arithmetic sequence. The common difference, 'd', for this sequence is 6. The first term, denoted by 'a₁' or simply 'a', is the first number in the sequence, which in this case is 25. Understanding these two key components – the first term and the common difference – is fundamental to deriving the general formula for the sequence.

The General Formula

The general formula for an arithmetic sequence is a powerful tool that allows us to find any term in the sequence without having to list out all the preceding terms. The formula is expressed as:

f(n) = a + (n - 1)d

Where:

  • f(n) represents the nth term in the sequence.
  • a represents the first term of the sequence.
  • n represents the position of the term in the sequence (e.g., 1 for the first term, 2 for the second term, and so on).
  • d represents the common difference between consecutive terms.

This formula essentially states that to find any term in the sequence, you start with the first term (a), and add the common difference (d) multiplied by one less than the term's position (n - 1). The (n - 1) factor accounts for the fact that the first term doesn't need the common difference added to it. It serves as the foundation for building our understanding of arithmetic sequences and forms the basis for finding a specific term within the series. This knowledge is crucial when working with more complex arithmetic sequence problems, and being able to apply the formula correctly is a valuable skill in mathematics.

Applying the Formula to Our Sequence

Now that we have the general formula and we know the first term (a = 25) and the common difference (d = 6) for our sequence, we can plug these values into the formula to find the specific formula for this sequence:

f(n) = 25 + (n - 1)6

This formula now represents the specific rule that governs our sequence 25, 31, 37, 43, 49... To ensure we fully grasp its implications, let's explore how this formula works for the first few terms. When n = 1 (the first term), f(1) = 25 + (1 - 1)6 = 25 + 0 = 25, which is correct. When n = 2 (the second term), f(2) = 25 + (2 - 1)6 = 25 + 6 = 31, also correct. For n = 3, f(3) = 25 + (3 - 1)6 = 25 + 12 = 37, which aligns perfectly with our sequence. This process of substituting values for 'n' reinforces our understanding of how the formula generates the terms of the sequence. The ability to derive this specific formula from the general formula is a key skill in working with arithmetic sequences, demonstrating a strong understanding of the underlying principles.

To further simplify our specific formula, we can distribute the 6 and combine like terms:

f(n) = 25 + 6n - 6

f(n) = 19 + 6n

This simplified form is equivalent to the previous one, just expressed in a slightly different way. It clearly shows the linear relationship between 'n' and the term value, highlighting the constant addition of the common difference. This simplified form will be useful when we compare it with the given options to identify the correct formula.

Analyzing the Options

Now, let's analyze the given options and compare them with the formula we derived (f(n) = 19 + 6n) to find the correct representation of the arithmetic sequence.

  • A. f(n) = 25 + 6(n)

    This option suggests that each term is obtained by adding 6 times the term number to the first term. Let's test it with n = 1: f(1) = 25 + 6(1) = 31. This is not the first term in our sequence, which is 25. Thus, option A is incorrect. The formula incorrectly calculates the first term and subsequent terms as well, diverging significantly from the actual sequence values.

  • B. f(n) = 25 + 6(n + 1)

    This option adds 1 to the term number before multiplying by the common difference. Let's test it with n = 1: f(1) = 25 + 6(1 + 1) = 25 + 12 = 37. This is not the first term (25), nor is it the second term (31). Hence, option B is also incorrect. The added 1 inside the parenthesis skews the calculation, leading to incorrect values for all terms in the sequence.

  • C. f(n) = 25 + 6(n - 1)

    This option aligns with the general formula we initially derived before simplifying. Let's test it with n = 1: f(1) = 25 + 6(1 - 1) = 25 + 0 = 25. This matches the first term. Let's test it with n = 2: f(2) = 25 + 6(2 - 1) = 25 + 6 = 31. This matches the second term. We can see that this formula correctly generates the terms of the sequence. It accurately reflects the fundamental structure of the arithmetic sequence, and is a strong candidate for the correct answer. It represents the initial formula before simplification and is a crucial step in finding the solution.

  • D. f(n) = 19 + 6(n + 1)

    This option uses 19 as a constant and adds 6 times (n + 1). Let's test it with n = 1: f(1) = 19 + 6(1 + 1) = 19 + 12 = 31. This is not the first term (25). Therefore, option D is incorrect. The combination of the constant 19 and the (n + 1) factor produces a formula that does not accurately represent the given sequence.

By carefully evaluating each option and comparing it with our derived formula and the actual sequence, we can confidently eliminate options A, B, and D. This process of elimination highlights the importance of meticulous analysis and the verification of each possible solution against the known characteristics of the sequence.

The Correct Answer

After analyzing all the options, we can conclude that the correct formula representing the arithmetic sequence 25, 31, 37, 43, 49... is:

C. f(n) = 25 + 6(n - 1)

This formula accurately captures the essence of the arithmetic sequence, using the first term (25) and the common difference (6) in the correct structure. It aligns perfectly with the general formula for arithmetic sequences and correctly generates each term in the given sequence. This comprehensive analysis demonstrates the power of understanding the fundamental concepts of arithmetic sequences and applying them to solve specific problems.

Conclusion

In this article, we embarked on a journey to understand and represent arithmetic sequences with formulas. We started by defining arithmetic sequences and identifying the first term and the common difference. We then explored the general formula for arithmetic sequences and applied it to our specific sequence 25, 31, 37, 43, 49.... We derived the formula f(n) = 25 + 6(n - 1) and simplified it to f(n) = 19 + 6n. Finally, we analyzed each option provided and confidently identified option C as the correct representation of the sequence. This process highlights the importance of understanding the underlying principles of arithmetic sequences and the systematic application of formulas to solve problems. By mastering these concepts, you can confidently tackle a wide range of arithmetic sequence challenges.