Question

If \(a_{1}, a_{2}, a_{3} \ldots\) are in HP and \(f(k)=\sum_{r=1}^{n} a_{r}-a_{k}\), then \(\frac{a_{1}}{f(1)}, \frac{a_{2}}{f(2)}, \frac{a_{3}}{f(3)}, \ldots \frac{a=}{f(n)}\) are in (a) AP (b) GP (c) HP (d) AGP

Short answer

Answer: The sequence is not in any of the given progressions (AP, GP, HP, or AGP).

Step-by-step solution

General term of HP

Let \(A\) be the first term of HP and \(d\) be the common difference of the corresponding AP. That is, \(a_1 = A\) and \(a_2 = A - d\). We know that in a harmonic progression, the reciprocals of the terms are in arithmetic progression. Therefore, we can write the general term of the given HP as follows: \[\frac{1}{a_k} = \frac{1}{A} + (k-1)\cdot\frac{1}{d}\]

Find the expression for f(k)

We are given the following expression for \(f(k)\): \[f(k) = \sum_{r=1}^n a_r - a_k\] Now, we need to find an expression for \(\sum_{r=1}^n a_r\). Using the general term of HP from Step 1, we find the sum: \[\sum_{r=1}^n a_r = \sum_{r=1}^n \frac{1}{\frac{1}{A} + (r-1)\cdot\frac{1}{d}} = \frac{d \cdot (1 + \ldots + n)}{n - d (1 + \ldots + n)}\] Substituting this in the expression for \(f(k)\), we get: \[f(k) = \frac{d \cdot (1 + \ldots + n)}{n - d (1 + \ldots + n)} - a_k\]

Analyze the sequence

Now, we have to study the sequence \(\frac{a_1}{f(1)}, \frac{a_2}{f(2)}, \frac{a_3}{f(3)}, \ldots \frac{a_n}{f(n)}\). Let's rewrite the terms using the expressions for \(a_k\) and \(f(k)\): \[\frac{a_k}{f(k)} = \frac{\frac{1}{\frac{1}{A} + (k-1)\cdot\frac{1}{d}}}{\frac{d \cdot (1 + \ldots + n)}{n - d (1 + \ldots + n)} - a_k}\] \[= \frac{a_k \cdot (n - d (1 + \ldots + n))}{d \cdot (1 + \ldots + n) - n \cdot a_k}\] Notice that the expression contains the terms \(1 + \ldots + n\) and \(a_k\). We can rewrite the expression as follows to see if it forms any of the given progressions: \[\frac{a_k}{f(k)} = \frac{a_k \cdot (n - kd)}{nd - n a_k}\] We can see from the expression that it doesn't form any AP, GP, or AGP. To check if it forms an HP, we can calculate the reciprocals of the terms. The reciprocal of the k-th term is given by: \[\frac{f(k)}{a_k} = \frac{nd - n a_k}{a_k \cdot (n - kd)}\] Comparing this expression with the general term of an HP, we can conclude that the given sequence is not in any of the given progressions.

Key concepts

Arithmetic Progression

When discussing harmonic progressions, arithmetic progressions (AP) play a central role because the reciprocal of a harmonic progression forms an arithmetic progression. An arithmetic sequence is a sequence of numbers such that the difference of any two successive members is a constant, called the common difference. For example, in the sequence 3, 6, 9, 12, the common difference is 3.
Understanding AP is crucial for analyzing HP since the reciprocal of an arithmetic progression leads to understanding the pattern and calculations of the harmonic progression.
In mathematical terms, if a sequence is arithmetic, the n-th term is given by the formula: \[a_n = a_1 + (n-1)d\] where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
By having a clear understanding of AP, students can delve deeper into sequences in mathematics, which serve as a foundation for many other progressions and series.

Sum of Series

A series is essentially the sum of the terms of a sequence. Calculating the sum of a series is often a crucial step in solving problems related to progressions like arithmetic or harmonic progression.
To find the sum of an arithmetic series, you can use the formula: \[S_n = \frac{n}{2} (a_1 + a_n)\] or alternatively, \[S_n = \frac{n}{2} (2a_1 + (n-1)d)\] where \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, and \(d\) is the common difference.
Understanding how to calculate the sum of a series not only helps appreciate the intricate pattern these numbers follow, but also aids in identifying and resolving problems associated with these sequences. It is particularly handy in proofs and other analytical applications in mathematics.

Sequences in Mathematics

Sequences are ordered lists of numbers following a specific pattern or rule. Each sequence has its own unique form and behavior, and understanding them is fundamental in many areas of mathematics.
Types of sequences include:

• Arithmetic Sequence: Each term is obtained by adding a constant to the previous term.
• Geometric Sequence: Each term is derived by multiplying the previous term by a constant.
• Harmonic Sequence: The reciprocal of the terms forms an arithmetic progression.
• Fibonacci Sequence: Each term is the sum of the two preceding ones.

The study of sequences allows mathematicians and students alike to model real-world situations and solve complex problems.
Understanding sequences is essential for anyone engaged in analytical thinking, as it forms the backbone of more advanced mathematical concepts like series, limits, and calculus.

Reciprocals in Progressions

Reciprocals in mathematical progressions involve taking the inverse of each term in a given sequence. This concept is especially important in harmonic progression, where the reciprocals of the terms give rise to an arithmetic sequence.
In simple terms, if you have a sequence in arithmetic progression, converting it into reciprocals will form a harmonic progression. For example, if the sequence in AP is \(2, 4, 6,\ldots\) its reciprocal will be \(\frac{1}{2}, \frac{1}{4}, \frac{1}{6},\ldots\), forming a harmonic sequence.
Understanding how taking reciprocals affect sequences can reveal new patterns and provide solutions to mathematical problems, including those that deal with time, rates, and speeds. This interplay highlights the rich interconnection between different types of sequences and how they can transform into one another through reciprocal relationships.