Finding The Nth Term Rule For The Sequence 3/4, 4/5, 5/6, 6/7

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Unveiling the Pattern in the Sequence

When presented with a sequence of numbers, our primary goal is to identify the underlying pattern that governs its progression. In this particular case, we are given the sequence 34,45,56,67,…\frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \ldots, and our mission is to determine the $n^{\text {th }} term rule, which will allow us to calculate any term in the sequence directly. To achieve this, we need to meticulously examine the relationship between the term number and the actual term value. Our main goal is to unravel the connection between the position of a term within the sequence and its corresponding numerical value. This involves a careful comparison of the numerators and denominators across the given terms, searching for a consistent pattern or formula that we can generalize. Think of it as solving a puzzle where each term is a clue, and the $n^{\text {th }} term rule is the final, elegant solution. We must analyze how the numbers in the sequence change from one term to the next.

Let's begin by scrutinizing the numerators and denominators separately. The numerators are 3, 4, 5, and 6, while the denominators are 4, 5, 6, and 7. Do you notice any immediate relationship between them? It seems that the denominator of each term is consistently one greater than its numerator. This is a crucial observation that will guide us in formulating the $n^{\text {th }} term rule. Now, we need to connect this observation to the term number, which represents the position of the term in the sequence. For example, the first term corresponds to $n = 1$, the second term to $n = 2$, and so on. Can we express the numerator and denominator in terms of $n$? This is the key question we need to answer to unlock the general formula for the sequence. Consider how the numerator changes as $n$ increases. When $n = 1$, the numerator is 3; when $n = 2$, the numerator is 4; and so forth. It appears that the numerator is always two more than the term number $n$. Similarly, the denominator seems to be three more than $n$. These initial observations are critical stepping stones towards finding the general rule. By breaking down the problem into smaller parts – analyzing numerators, denominators, and their relationship to the term number – we are employing a methodical approach to problem-solving. This strategy is not only useful in mathematics but also in many other areas of life where complex problems need to be tackled. The ability to break down a complex problem into smaller, more manageable components is a hallmark of effective problem-solving. This allows us to focus on individual aspects of the problem, identify patterns, and gradually build towards a comprehensive solution.

Deconstructing the Numerator and Denominator Relationship

To formalize our findings, let's delve deeper into the connection between the numerator, denominator, and the term number $n$. We noticed that the numerator of each fraction is two more than the term number. Mathematically, we can express this as numerator = $n + 2$. Similarly, the denominator is three more than the term number, which can be written as denominator = $n + 3$. Now, we have expressions for both the numerator and the denominator in terms of $n$. The next step is to combine these expressions to form the general formula for the $n^{\text {th }} term of the sequence. Remember, our ultimate goal is to express the entire fraction (the $n^{\text {th }} term) as a single expression involving $n$. This will allow us to calculate any term in the sequence, no matter how far down the line it is. Consider how this approach highlights the power of algebraic representation in mathematics. By using variables like $n$, we can generalize patterns and relationships, allowing us to work with sequences and other mathematical concepts in a much more flexible and efficient way. The ability to express mathematical ideas using symbols and equations is a fundamental skill that opens up a vast world of mathematical possibilities. It is a language that allows us to communicate complex ideas concisely and precisely.

Let's take a moment to verify our expressions for the numerator and denominator against the given sequence. For the first term ($n = 1$), our expressions give us a numerator of $1 + 2 = 3$ and a denominator of $1 + 3 = 4$, which matches the first term $\frac{3}{4}$. For the second term ($n = 2$), we get a numerator of $2 + 2 = 4$ and a denominator of $2 + 3 = 5$, corresponding to the term $\frac{4}{5}$. This confirmation strengthens our confidence in the expressions we have derived. This process of verification is a crucial step in any mathematical problem-solving endeavor. It allows us to catch any errors or inconsistencies early on and ensures that our solution is accurate. By testing our expressions against the known terms of the sequence, we are essentially performing a reality check, making sure that our mathematical model aligns with the observed data. This iterative process of formulating a hypothesis, testing it, and refining it based on the results is a core principle of the scientific method and is equally applicable in mathematics. Now that we have expressions for the numerator and denominator in terms of $n$, we are ready to combine them to form the general formula for the $n^{\text {th }} term. This will be the culmination of our analysis, providing us with a powerful tool for understanding and working with the sequence.

Formulating the $n^{\text {th }} Term Rule

Having established that the numerator is $n + 2$ and the denominator is $n + 3$, we can now express the $n^{\text {th }} term of the sequence as a fraction. The $n^{\text {th }} term is simply the numerator divided by the denominator, which gives us $\frac{n + 2}{n + 3}$. This is the general formula for the sequence, and it encapsulates the pattern we observed in the given terms. This elegant formula is the culmination of our efforts to understand the sequence. It is a compact and powerful representation of the underlying pattern that governs the sequence's progression. With this formula in hand, we can now calculate any term in the sequence without having to list out all the preceding terms. This is the true power of mathematical generalization – the ability to express a specific pattern in a way that allows us to make predictions and calculations for an infinite number of cases. Consider the implications of this formula. It allows us to not only find the next few terms in the sequence but also to calculate terms that are far down the line, such as the 100th term or even the 1000th term. This demonstrates the predictive power of mathematical formulas and their ability to extend our understanding beyond the immediate data we are given.

To ensure our formula is correct and in its simplest form, we need to consider whether the fraction $\frac{n + 2}{n + 3}$ can be simplified further. In this case, $n + 2$ and $n + 3$ are consecutive integers, meaning they share no common factors other than 1. Therefore, the fraction is already in its simplest form. This is an important consideration when working with fractions. We always want to express our answer in the simplest possible form, which means reducing the fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor. In this case, since $n + 2$ and $n + 3$ are consecutive integers, they are relatively prime, meaning their greatest common divisor is 1. Therefore, the fraction cannot be simplified any further. The fact that the fraction is already in its simplest form adds to the elegance of the solution. It means that the formula we have derived is not only accurate but also as concise as possible. This is a desirable characteristic of mathematical solutions – we strive for both accuracy and simplicity. A solution that is both correct and easy to understand is generally considered more valuable than a solution that is correct but complex and difficult to interpret. Now that we have our final answer, we can confidently state the $n^{\text {th }} term rule for the sequence.

The $n^{\text {th }} Term Rule: A Concise Summary

In conclusion, the $n^{\text {th }} term rule for the sequence 34,45,56,67,…\frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \ldots is given by the fraction $\frac{n + 2}{n + 3}$. This formula elegantly captures the pattern observed in the sequence, where each term's numerator is two more than its position in the sequence, and the denominator is three more than its position. The fraction is already in its simplest form, as the numerator and denominator are consecutive integers and share no common factors other than 1. This result demonstrates the power of careful observation and algebraic manipulation in uncovering underlying patterns in sequences. By systematically analyzing the relationship between the term number and the term value, we were able to derive a concise and accurate formula that allows us to calculate any term in the sequence. This process highlights the beauty and elegance of mathematics, where complex patterns can be expressed in simple and meaningful ways. The ability to identify patterns and express them mathematically is a valuable skill that can be applied in various fields, from science and engineering to finance and computer science. This problem-solving approach, which involves breaking down a complex problem into smaller, manageable steps, is a universally applicable skill that extends beyond the realm of mathematics. It is a valuable tool for tackling challenges in any area of life.