Question

The 10 th term of harmonic progression \(\frac{1}{5}, \frac{4}{19}, \frac{2}{9}, \frac{4}{17}, \ldots \ldots \ldots\) is (1) \(\frac{11}{4}\) (2) \(\frac{13}{4}\) (3) \(\frac{4}{13}\) (4) \(\frac{4}{11}\)

Short answer

Solution: Upon analyzing the reciprocals of the given terms and comparing their differences, we found three values (\(d_1 = \frac{15}{4}\), \(d_2 = \frac{5}{4}\), and \(d_3 = \frac{-3}{4}\)) as the common difference, proving that the sequence isn't a harmonic progression. Therefore, there's no possible solution for the 10th term, and the given options are incorrect.

Step-by-step solution

Find the common difference

Recall that a sequence is a harmonic progression if its reciprocals form an arithmetic progression. So, to find the common difference, let's first find the reciprocals of the given terms. Reciprocal of \(\frac{1}{5}\) is \(5\) Reciprocal of \(\frac{4}{19}\) is \(\frac{19}{4}\) Reciprocal of \(\frac{2}{9}\) is \(\frac{9}{2}\) Reciprocal of \(\frac{4}{17}\) is \(\frac{17}{4}\) Now let's find the common difference in this arithmetic progression. \(d_1 =\frac{19}{4} - 5 = \frac{15}{4}\) \(d_2 = \frac{9}{2} -\frac{19}{4} = \frac{5}{4}\) \(d_3 = \frac{17}{4} - \frac{9}{2} = \frac{-3}{4}\) We have different common differences here, it indicates the given progression is not harmonic. So, there is no possible solution for the 10th term. Hence, the given options are incorrect, and the progression is not a harmonic progression.

Key concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is always the same, which is called the common difference. This type of sequence is not only foundational in mathematics but also easy to identify once you know how it works.
For example, in the sequence 2, 4, 6, 8, the common difference is 2, since each term increases by the same amount. Similarly, in a decreasing sequence like 10, 7, 4, 1, the common difference would be -3, as each term is reduced by 3.
To identify the common difference, you subtract any term in the sequence from the term that follows it. If the result is the same for all pairs of consecutive terms, you are dealing with an arithmetic progression. It's as simple as that:

• Term 2 - Term 1 = Common Difference
• Term 3 - Term 2 = Common Difference
• ... and so forth for all terms in the sequence

Reciprocals

Reciprocals are fundamental in understanding sequences, especially in distinguishing different types of progressions like harmonic ones. The reciprocal of a number is simply 1 divided by that number. So, for a number \(x\), its reciprocal is \(\frac{1}{x}\).
For example, the reciprocal of 5 is \(\frac{1}{5}\), and the reciprocal of \(\frac{3}{4}\) would be \(\frac{4}{3}\). When dealing with sequences, putting terms into reciprocals can often help in identifying if they form a particular type of progression, such as an arithmetic progression.
In our original problem, the given terms in the sequence were checked if their reciprocals form an arithmetic progression, crucial for identifying if the sequence is a harmonic progression. However, only once their reciprocals form an arithmetic sequence does it indicate that the original sequence is harmonic.

Common Difference

The common difference is the fixed amount added to each term of an arithmetic progression to get the next term. It gives the AP its structure and is integral in predicting future terms. This concept lets you describe the entire sequence with a starting term and this common difference.
In arithmetic progressions, the common difference is calculated by: \[d = a_{n} - a_{n-1}\]where \(a_{n}\) refers to the nth term of the sequence.
In harmonic progressions, identifying a common difference is actually about finding the common difference of the sequence formed by the reciprocals of the terms. If there's inconsistency in these differences, as seen in the exercise, it indicates that the sequence isn't harmonic. The value of the common difference essentially encapsulates the change happening across the sequence from term to term.

Sequence Analysis

Sequence analysis refers to the study and understanding of the arrangement of numbers in a sequence, and how it behaves over increasing terms. It involves exploring the types of sequences, their properties, and the underlying rules that govern them.
This practice is crucial in mathematics to predict subsequent terms or to identify patterns. For example, sequence analysis can help determine if a given sequence is arithmetic, geometric, harmonic, or something else entirely.
In the original exercise, sequence analysis was applied to ascertain whether the sequence of reciprocals followed an arithmetic progression, which would confirm the original to be harmonic. Upon analysis, the inconsistencies in the common differences showed that the sequence didn't follow harmonic rules, leading to the conclusion that it wasn't a harmonic sequence despite its initial appearance.