Finding The Explicit Formula For The Arithmetic Sequence 4/5, 29/30, 17/15, 13/10

by ADMIN 82 views

Delving into the world of arithmetic sequences, we encounter a fascinating pattern where each term progresses by a constant difference. In this article, we will embark on a journey to decipher the explicit formula for the given arithmetic sequence: 4/5, 29/30, 17/15, 13/10. Our mission is to unearth the rule that governs this sequence, allowing us to predict any term in the series with ease. Let's explore the intricacies of arithmetic sequences and derive the explicit formula that unlocks the secrets of this particular progression.

Understanding Arithmetic Sequences

Before we dive into the specifics of our sequence, let's lay the groundwork by understanding the fundamental principles of arithmetic sequences. An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms remains constant. This constant difference is known as the common difference, often denoted by 'd'.

To illustrate, consider the sequence 2, 5, 8, 11, 14... Here, the common difference 'd' is 3, as each term is obtained by adding 3 to the previous term. This consistent pattern of adding the same value to generate subsequent terms is the hallmark of an arithmetic sequence.

The general form of an arithmetic sequence can be expressed as:

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

Where:

  • 'a' represents the first term of the sequence.
  • 'd' represents the common difference.

Key Characteristics of Arithmetic Sequences:

  • Constant Difference: The defining trait of an arithmetic sequence is the constant difference between consecutive terms. This uniformity allows for predictable progression within the sequence.
  • Linear Progression: Arithmetic sequences exhibit a linear progression, meaning the terms increase or decrease at a constant rate. This linear nature makes them amenable to algebraic representation.
  • Explicit Formula: A powerful tool for working with arithmetic sequences is the explicit formula, which provides a direct way to calculate any term in the sequence without having to list out all the preceding terms. We will delve into this formula shortly.

The Explicit Formula: A Gateway to Any Term

The explicit formula is a cornerstone in the study of arithmetic sequences. It acts as a mathematical portal, allowing us to determine the value of any term in the sequence directly, given its position. This formula eliminates the need to recursively calculate terms, making it an invaluable tool for analyzing and predicting the behavior of arithmetic sequences.

The explicit formula for an arithmetic sequence is given by:

a_n = a_1 + (n - 1)d

Where:

  • a_n represents the nth term of the sequence (the term we want to find).
  • a_1 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.

Dissecting the Formula:

The formula encapsulates the essence of an arithmetic sequence's progression. Let's break it down to understand its components:

  • a_1: The starting point. It's the initial value upon which the sequence is built.
  • (n - 1): This term reflects the number of 'jumps' or common differences needed to reach the nth term from the first term. For instance, to reach the 5th term, we need to add the common difference 4 times (5 - 1 = 4).
  • d: The constant increment. It's the value added to each term to generate the next term in the sequence.

Applying the Explicit Formula: A Practical Example

To solidify our understanding, let's consider a simple arithmetic sequence: 3, 7, 11, 15, ...

Here,

  • a_1 = 3 (the first term)
  • d = 4 (the common difference)

Suppose we want to find the 10th term (a_10). Using the explicit formula:

a_10 = a_1 + (10 - 1)d
a_10 = 3 + (9)4
a_10 = 3 + 36
a_10 = 39

Therefore, the 10th term in the sequence is 39. This example demonstrates the power of the explicit formula in efficiently determining any term in an arithmetic sequence.

Decoding the Sequence: 4/5, 29/30, 17/15, 13/10

Now, let's turn our attention to the sequence in question: 4/5, 29/30, 17/15, 13/10. Our objective is to determine the explicit formula that governs this particular sequence. To do so, we need to identify the first term (a_1) and the common difference (d).

Identifying the First Term (a_1):

The first term of the sequence is simply the initial value, which is 4/5. So, we have a_1 = 4/5.

Calculating the Common Difference (d):

To find the common difference, we subtract any term from its succeeding term. Let's subtract the first term from the second term:

d = (29/30) - (4/5)

To perform this subtraction, we need a common denominator. The least common denominator for 30 and 5 is 30. So, we rewrite 4/5 as 24/30:

d = (29/30) - (24/30)
d = 5/30
d = 1/6

Therefore, the common difference d is 1/6. It's crucial to verify this common difference by checking if it holds true for other consecutive terms in the sequence. Let's subtract the second term from the third term:

(17/15) - (29/30)

Again, we need a common denominator. The least common denominator for 15 and 30 is 30. So, we rewrite 17/15 as 34/30:

(34/30) - (29/30) = 5/30 = 1/6

The common difference remains consistent, confirming that our sequence is indeed arithmetic.

Constructing the Explicit Formula

With the first term (a_1 = 4/5) and the common difference (d = 1/6) in hand, we can now construct the explicit formula for the sequence. We plug these values into the general explicit formula:

a_n = a_1 + (n - 1)d
a_n = (4/5) + (n - 1)(1/6)

This formula provides a direct route to calculate any term (a_n) in the sequence, given its position (n).

Evaluating the Answer Choices

Now that we have derived the explicit formula, let's compare it to the answer choices provided to identify the correct option:

A. a_n = -1/6 + 4/5(n-1) B. a_n = 1/6 ÷ 4/5(n-1) C. a_n = 4/5 + (-1/6)(n-1)

Our derived explicit formula is:

a_n = (4/5) + (n - 1)(1/6)

By comparing, we can see that option C closely resembles our derived formula. However, there's a slight difference in the sign of the common difference term. Option C has a negative common difference (-1/6), while our calculated common difference is positive (1/6). This discrepancy indicates that option C is incorrect.

Let's analyze option A. It presents the formula as:

a_n = -1/6 + 4/5(n-1)

This formula seems to have the first term and the common difference interchanged and also has a negative sign in the first term. Let's rearrange our derived formula to see if it can match option A:

a_n = (4/5) + (1/6)(n - 1)
a_n = (4/5) + (1/6)n - (1/6)
a_n = (1/6)n + (4/5) - (1/6)

Now, let's find a common denominator for 4/5 and 1/6, which is 30:

a_n = (1/6)n + (24/30) - (5/30)
a_n = (1/6)n + (19/30)

This form doesn't directly match option A. However, let's manipulate option A to see if we can make a comparison:

a_n = -1/6 + (4/5)(n - 1)
a_n = -1/6 + (4/5)n - (4/5)

Finding a common denominator of 30:

a_n = -(5/30) + (24/30)n - (24/30)
a_n = (24/30)n - (29/30)

This also doesn't match our derived formula. Therefore, option A is incorrect.

Option B presents a different structure altogether, involving division. This form is not consistent with the standard explicit formula for arithmetic sequences. Therefore, option B is also incorrect.

The Correct Explicit Formula

After careful analysis, we've identified that none of the provided options perfectly match the explicit formula we derived. This highlights the importance of understanding the underlying concepts and deriving the formula independently. Let's restate the correct explicit formula for the arithmetic sequence 4/5, 29/30, 17/15, 13/10:

a_n = (4/5) + (1/6)(n - 1)

This formula accurately represents the sequence, allowing us to calculate any term in the series. For instance, to find the 5th term, we substitute n = 5:

a_5 = (4/5) + (1/6)(5 - 1)
a_5 = (4/5) + (1/6)(4)
a_5 = (4/5) + (4/6)
a_5 = (4/5) + (2/3)

Finding a common denominator of 15:

a_5 = (12/15) + (10/15)
a_5 = 22/15

Thus, the 5th term in the sequence is 22/15. This further validates our derived explicit formula.

Conclusion

In this exploration, we've successfully unveiled the explicit formula for the arithmetic sequence 4/5, 29/30, 17/15, 13/10. We began by establishing a firm understanding of arithmetic sequences and the significance of the explicit formula. We then meticulously calculated the first term and the common difference of the given sequence. By plugging these values into the general explicit formula, we derived the specific formula for this sequence:

a_n = (4/5) + (1/6)(n - 1)

We also critically evaluated the provided answer choices, demonstrating the importance of independently deriving the formula to ensure accuracy. This journey through arithmetic sequences has underscored the power of explicit formulas in predicting and analyzing patterns in mathematical progressions. Understanding these concepts empowers us to tackle a wide range of sequence-related problems with confidence and precision.

Key Takeaways:

  • Arithmetic sequences exhibit a constant difference between consecutive terms.
  • The explicit formula provides a direct way to calculate any term in an arithmetic sequence.
  • Deriving the formula independently ensures accuracy and a deeper understanding of the sequence.
  • The explicit formula for the sequence 4/5, 29/30, 17/15, 13/10 is a_n = (4/5) + (1/6)(n - 1).

This exploration has not only provided the solution to the specific problem but has also reinforced the fundamental principles of arithmetic sequences and the power of the explicit formula. With this knowledge, we are well-equipped to tackle future challenges in the realm of mathematical sequences and patterns.