Problem 113 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(a, b, c\) are in AP, \(b-a, c-b\) and \(a\) are in GP, then find \(a: b: c\).

Step-by-Step Solution

Answer

The ratio of \(a:b:c\) is 1:1:3.

Step 1: Express Variables in the Form of Arithmetic Progression

Since \(a, b, c\) are in AP, we can express \(b\) as the average of \(a\) and \(c\), or \(b = \frac{a+c}{2}\).

Step 2: Express Variables in the Geometric Progression Form and Find the Common Ratio

Since \(b - a\), \(c - b\) and \(a\) are in GP, we can say that \(\frac{c - b}{b - a} = \frac{b - a}{a}\). This equation is derived from the fact that the ratio between successive terms should be equal in a geometric progression.

Step 3: Simplify the Equation to Find the Ratio

Substitute \(b - a\) with \(c - 2b + a\) in the equation from Step 2 and simplify to get \(c = 3b\). Now we substitute \(b = \frac{a+c}{2}\) from Step 1 and \(c = 3b\) into each other and simplify to get \(a = b\). So we got \(a = b\) and \(c = 3b\). Therefore, the ratio is \(a:b:c = 1:1:3\).

Key Concepts

Arithmetic Progression

Arithmetic progression (AP) is a sequence of numbers in which the difference between the consecutive terms is constant. This difference is commonly known as the common difference. In the given problem, we have three terms, a, b, and c, which are in AP. To visualize this, consider that you are climbing stairs, and every step you take is of the same height—that step is analogous to the common difference in AP.

An important property of AP is that any term can be expressed using others. For example, the middle term b can indeed be represented as the average of a and c because it lies exactly in the middle of these two terms in the sequence. Mathematically, this is b = \( \frac{a+c}{2} \). This idea is fundamental to solving problems involving AP, like in our exercise.

Geometric Progression

Geometric progression (GP), unlike the arithmetic progression, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. Think of it like a chain reaction where you start with a single entity, and each subsequent entity is k times larger or smaller based on this common ratio.

For the terms (b-a), (c-b), and a to be in GP, as per the problem, they must satisfy the condition that the ratio of any term to its preceding term is constant. This is expressed as: \( \frac{c - b}{b - a} = \frac{b - a}{a} \). This equation is crucial because it allows us to relate the terms and ultimately determine the actual values of a, b, and c that meet the conditions of both AP and GP—the crux of our exercise.

Sequence and Series

A sequence is an ordered list of numbers following some well-defined rule, such as a, b, and c from our exercise. When the terms of a sequence are summed up, it becomes a series. Understanding the nature of sequences and series is vital to tackle various mathematical problems, particularly in algebra and calculus.

To decipher sequences, we often look for patterns. For AP and GP, these patterns are the common difference and the common ratio, respectively. Conceptualizing the terms of a sequence in real-world situations, like timing of a clock or payment installments, can aid in comprehending the abstract mathematical concepts.

In the context of our problem, the relationship between the terms reaches beyond simple addition or multiplication; it involves discerning patterns and applying them to link AP and GP, a common challenge in sequence and series problems.

IIT JEE Mathematics

The Indian Institute of Technology Joint Entrance Examination (IIT JEE) is an academically challenging test that assesses a student's aptitude in various subjects, including Mathematics. This problem is a quintessential example of the complexity involved in IIT JEE Mathematics—a segment revered for its rigorous and thought-provoking questions covering a broad spectrum of topics like sequences and series, calculus, and algebra.

Success in this exam hinges not only on understanding the fundamental concepts of arithmetic and geometric progressions but also on an ability to relate disparate ideas. Coming across exercises such as finding the ratio of terms which are simultaneously part of an AP and a GP, encourages students to apply a blend of analytical skills and mathematical knowledge—an intersection crucial for conquering IIT JEE mathematics puzzles.