Let \(a_{1}, a_{2}, a_{3}, \ldots\) be in harmonic progression with \(a_{1}=5\) and \(a_{20}=25 .\) The least positive integer \(n\) for which \(a_{n}<0\) is (A) 22 (B) 23 (C) 24 (D) 25

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between the consecutive terms is constant. This difference is known as the common difference. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2. Arithmetic sequences are often represented by terms like a₁, a₂, a₃, and so on, where a_n denotes the nth term.

In practical situations, APs can describe scenarios like the regular addition of money into a savings account or evenly increasing the dose of medication. For students preparing for competitive exams like JEE Advanced, understanding APs is fundamental, as it's an integral part of higher secondary mathematics syllabi and often appears in various problems. A strong grasp of AP concepts not only helps in direct problems but also ones involving complex numbers, calculus, and even physics inquiries related to motion.

The common difference is the consistent interval or space between terms in an arithmetic progression. It is denoted by d and can be positive, negative, or zero, affecting the sequence's nature. If d is positive, the AP terms increase, making the sequence ascending. If d is negative, the terms decrease, resulting in a descending sequence. For a zero common difference, all terms are the same, which means it's essentially a constant sequence.

The common difference is calculated as d = a_n - a_n-1, where a_n and a_n-1 are subsequent terms in the progression. Knowing the common difference is essential for solving many problems in JEE mathematics and is a stepping stone to understanding more complex series and sequences.

JEE Advanced mathematics tests students on their analytical and conceptual understanding of various high-school level mathematics topics. It includes calculus, algebra, geometry, and trigonometry. Sequences and series form a critical part of JEE Advanced syllabus, and proficiency in these is crucial. Specifically, problems related to Arithmetic and Harmonic Progressions (AP and HP) often feature in the exam. A deep understanding of these topics not only helps in solving direct questions but also in tactfully handling complex real-world problems. JEE questions are designed to test conceptual clarity, speed, and application, which requires students to be thorough with formulas, quick in calculations, and sharp in applying multiple concepts to solve a single problem.

The exercise provided illustrates an example of how harmonic progression is applied by converting it to an arithmetic progression to find the solution—a skill that can be advantageous in the JEE Advanced test.

The nth term of an arithmetic progression can be found using the formula a_n = a₁ + (n-1)d, where a₁ is the first term, n is the term number, and d is the common difference. This formula allows us to find any term in the sequence without having to list all preceding terms, which is especially useful in dealing with large values of n.

For instance, if we need to determine the 100th term of the AP 3, 7, 11, ..., we know the first term a₁ = 3 and the common difference d = 4. Using the nth term formula, a₁₀₀ = a₁ + (100-1)d = 3 + 99*4 = 399. This concept is pivotal in various calculations and forms the basis for solving sequence-related problems, such as the exercise given, which involves understanding the nth term in the context of harmonic progressions converted to arithmetic progressions.