Finding The 15th Term Of The Harmonic Sequence 1/3, 1/6, 1/9, ...
This article provides a detailed explanation on how to find the 15th term of the given harmonic sequence: 1/3, 1/6, 1/9, ... We will first explore the fundamentals of harmonic sequences and their relationship with arithmetic sequences. Then, we will walk through the step-by-step process of identifying the underlying arithmetic sequence, calculating the nth term, and finally, determining the 15th term of the original harmonic sequence. Whether you're a student tackling sequence problems or simply curious about mathematical patterns, this guide will equip you with the knowledge to solve similar problems.
Understanding Harmonic Sequences
Harmonic sequences are a fascinating type of sequence in mathematics, closely related to arithmetic sequences. To truly grasp harmonic sequences, it's essential to first understand their connection to arithmetic sequences. An arithmetic sequence is a sequence of numbers where the difference between consecutive terms remains constant. This constant difference is known as the common difference. For instance, the sequence 2, 4, 6, 8, ... is an arithmetic sequence with a common difference of 2. Now, a harmonic sequence is formed by taking the reciprocals of the terms in an arithmetic sequence. This means that if you have an arithmetic sequence, you can create a harmonic sequence by simply inverting each term (i.e., taking 1 divided by each term). This inverse relationship is the cornerstone of understanding and working with harmonic sequences. Identifying a harmonic sequence is straightforward: If the reciprocals of the terms form an arithmetic sequence, then the original sequence is harmonic. This simple test is key to unlocking the properties and patterns within harmonic sequences.
Consider the given sequence: 1/3, 1/6, 1/9, ... To determine if it's a harmonic sequence, we need to examine the reciprocals of its terms. The reciprocals are 3, 6, 9, ... This new sequence appears to be an arithmetic sequence. To confirm, we check the difference between consecutive terms. The difference between 6 and 3 is 3, and the difference between 9 and 6 is also 3. Since the difference is constant, the sequence 3, 6, 9, ... is indeed an arithmetic sequence. Therefore, the original sequence, 1/3, 1/6, 1/9, ... is a harmonic sequence. This initial step of confirming the harmonic nature of the sequence is crucial because it allows us to leverage the properties of arithmetic sequences to solve for terms in the harmonic sequence. Understanding this fundamental relationship is key to navigating and solving problems involving harmonic sequences effectively.
Finding the Corresponding Arithmetic Sequence
To find the corresponding arithmetic sequence for the given harmonic sequence 1/3, 1/6, 1/9, ..., we take the reciprocals of each term. This process transforms the harmonic sequence into its associated arithmetic sequence, which is much easier to work with when finding specific terms. By taking the reciprocal of 1/3, we get 3. Similarly, the reciprocal of 1/6 is 6, and the reciprocal of 1/9 is 9. Therefore, the corresponding arithmetic sequence is 3, 6, 9, ... Now that we have the arithmetic sequence, we need to identify its key characteristics: the first term and the common difference. These two values are essential for determining any term in the arithmetic sequence, and subsequently, in the harmonic sequence. The first term of the arithmetic sequence is simply the first number in the sequence, which is 3 in this case. This is our starting point. Next, we need to find the common difference. The common difference is the constant value added to each term to get the next term in the sequence. We can find it by subtracting any term from its subsequent term. For example, subtracting the first term (3) from the second term (6) gives us 6 - 3 = 3. We can verify this by subtracting the second term (6) from the third term (9), which also gives us 9 - 6 = 3. Thus, the common difference of this arithmetic sequence is 3. With the first term (3) and the common difference (3) identified, we have all the information needed to determine any term in this arithmetic sequence. This is a critical step in solving the problem because once we find the nth term of the arithmetic sequence, we can easily find the nth term of the harmonic sequence by taking its reciprocal. The clear connection between the harmonic and arithmetic sequences makes this method a powerful tool for solving such problems.
Determining the nth Term of the Arithmetic Sequence
To determine the nth term of the arithmetic sequence, we utilize the well-established formula for the nth term of an arithmetic sequence. This formula provides a direct method for calculating any term in the sequence, given the first term, the common difference, and the term number (n). The formula is expressed as: a_n = a_1 + (n - 1)d, where a_n represents the nth term, a_1 represents the first term, n is the term number, and d is the common difference. This formula is a cornerstone in working with arithmetic sequences and provides a systematic approach to finding any term without having to list out all the preceding terms. Applying this formula to our sequence, we have already identified a_1 as 3 (the first term) and d as 3 (the common difference). We are interested in finding the 15th term, so n = 15. Substituting these values into the formula, we get: a_15 = 3 + (15 - 1) * 3. Now, we simplify the expression step-by-step. First, we calculate the value inside the parentheses: 15 - 1 = 14. Next, we multiply this result by the common difference: 14 * 3 = 42. Finally, we add this to the first term: 3 + 42 = 45. Therefore, the 15th term (a_15) of the arithmetic sequence is 45. This calculation is a crucial intermediate step, as it provides us with the value needed to find the 15th term of the original harmonic sequence. The formula allows us to jump directly to the term we need without needing to calculate all the terms leading up to it, saving time and reducing the chance of error. Understanding and applying this formula effectively is a key skill in solving problems involving arithmetic and harmonic sequences.
Finding the 15th Term of the Harmonic Sequence
Having found the 15th term of the corresponding arithmetic sequence, which is 45, we are now in the final stage of determining the 15th term of the original harmonic sequence. The relationship between harmonic and arithmetic sequences is based on reciprocals, meaning that if we have the nth term of the arithmetic sequence, we can find the nth term of the harmonic sequence by simply taking its reciprocal. In this case, we found that the 15th term of the arithmetic sequence (3, 6, 9, ...) is 45. Therefore, to find the 15th term of the harmonic sequence (1/3, 1/6, 1/9, ...), we take the reciprocal of 45. The reciprocal of 45 is 1/45. Thus, the 15th term of the given harmonic sequence is 1/45. This final step demonstrates the elegance and simplicity of working with harmonic sequences by utilizing their connection to arithmetic sequences. By converting the harmonic sequence to its corresponding arithmetic sequence, applying the formula for the nth term, and then taking the reciprocal, we efficiently found the desired term. This process highlights the importance of understanding the relationships between different types of sequences in mathematics. Understanding this reciprocal relationship is key to solving harmonic sequence problems effectively and accurately.
Therefore, the 15th term of the harmonic sequence 1/3, 1/6, 1/9, ... is 1/45.
