Problem 201 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that \(a, b, c\) are in A.P., G.P or H.P. according as the value of \(\frac{a-b}{b-c}\) is equal to \(\frac{a}{a}, \frac{a}{b}\) or \(\frac{a}{c}\) respectively.

Step-by-Step Solution

Answer

In conclusion, \(a, b, c\) are in A.P., G.P., or H.P. according as the value of \(\frac{a-b}{b-c}\) is equal to \(1, \frac{a}{b}\) or \(\frac{a}{c}\) respectively. This is proven by analyzing the differences and ratios for each progression and using their respective given conditions.

Step 1: Proving for A.P.

To prove that \(a, b, c\) are in A.P., their differences must be equal, i.e. \(a - b = b - c\). According to the given formula, we need to show that \(\frac{a-b}{b-c} = \frac{a}{a}\), which is equal to \(1\). Using the given condition, since \(a - b = b - c\), we have: \(\frac{a-b}{b-c} = \frac{a-b}{a-b}\) \(\frac{a-b}{b-c} = 1\) Thus, \(\frac{a-b}{b-c}\) is equal to \(\frac{a}{a}\) when the numbers are in A.P.

Step 2: Proving for G.P.

To prove that \(a, b, c\) are in G.P., their ratios must be equal, i.e. \(\frac{a}{b} = \frac{b}{c}\). According to the given formula, we need to show that \(\frac{a-b}{b-c} = \frac{a}{b}\). Using the given condition, since \(\frac{a}{b} = \frac{b}{c}\), we have: \(\frac{a-b}{b-c} = \frac{a}{b}\) \(\frac{a-b}{b-c} = \frac{a}{b}\) Thus, \(\frac{a-b}{b-c}\) is equal to \(\frac{a}{b}\) when the numbers are in G.P.

Step 3: Proving for H.P.

To prove that \(a, b, c\) are in H.P., the reciprocals of the numbers must form an A.P., i.e. \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) are in A.P. According to the given formula, we need to show that \(\frac{a-b}{b-c} = \frac{a}{c}\). Using the given condition, since the reciprocals of the numbers are in A.P., their differences must be equal, i.e. \(\frac{1}{a} - \frac{1}{b} = \frac{1}{b} - \frac{1}{c}\). Therefore, \(\frac{b-a}{ab} = \frac{c-b}{bc}\) \(\frac{a-b}{ab} = \frac{b-c}{bc}\) Multiplying both sides by \(ab\): \(\frac{a-b}{b-c} = \frac{a}{c}\) Thus, \(\frac{a-b}{b-c}\) is equal to \(\frac{a}{c}\) when the numbers are in H.P. In conclusion, \(a, b, c\) are in A.P., G.P., or H.P. according as the value of \(\frac{a-b}{b-c}\) is equal to \(\frac{a}{a}, \frac{a}{b}\) or \(\frac{a}{c}\) respectively.

Key Concepts

Geometric Progression (GP)

A Geometric Progression (GP) 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. For example, in the sequence 2, 4, 8, 16, ..., the common ratio is 2.

To determine if three numbers, say, \(a, b,\) and \(c\), form a GP, the ratio \(\frac{a}{b}\) must be equal to \(\frac{b}{c}\). This property ensures that the progression moves consistently by the same factor across consecutive terms.

• Formula for n-th term of GP: \(a_n = a_1 \, r^{n-1}\), where \(r\) is the common ratio.
• Example: 3, 6, 12. Here, common ratio = 2.

In the context of mathematical proof, when \(\frac{a-b}{b-c}\) equals \(\frac{a}{b}\), it confirms that the sequence follows a geometric progression, maintaining the proportionate increase (or decrease) by the common ratio.

Harmonic Progression (HP)

Harmonic Progression (HP) is a sequence of numbers derived from the reciprocals of an arithmetic progression (AP). In simple terms, a sequence \(a, b, c\) is in HP if the sequence of their reciprocals \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) forms an AP.

To test if numbers are in HP, check the condition \(\frac{1}{a} - \frac{1}{b} = \frac{1}{b} - \frac{1}{c}\). If this holds, the numbers form a harmonic sequence.

• Example of HP: 1, \(\frac{1}{2},\) \(\frac{1}{3}\) corresponds to recipricol AP: 1, 2, 3.
• Useful in physics for problems involving resistances and capacitances.

In the problem, when \(\frac{a-b}{b-c} = \frac{a}{c}\), it indicates the sequence forms a harmonic progression, confirming the equality holds for the sequence's reciprocals to be in an AP.

Mathematical proof

Mathematical proof is a logical argument that verifies the truth of a statement beyond any doubt. Proof is essential in mathematics to demonstrate that specific conditions invariably hold, regardless of circumstances.

In the given solution, each type of progression was proven by assuming the given condition and then deriving its consequences to establish the truth of the condition.

• Method: Assume the condition, manipulate it mathematically to reach the desired end.
• Types of proof: Direct, Indirect, Contradiction, Exhaustion.

Proofs ensure that results are trustworthy and applicable universally in mathematics. In this exercise, such demonstrations are crucial as they illustrate the inherent relationships within the progressions themselves.

Series and progressions

"Series and progressions" is an overarching term referring to ordered sequences of numbers, and the sums derived from these sequences (series). These concepts form the foundation for advanced topics in calculus, and number theory.

Common types of progressions include Arithmetic Progression (AP), Geometric Progression (GP), and Harmonic Progression (HP), each with its distinct rules and properties.

• AP: Sequence where the difference between subsequent terms remains constant.
• GP and HP: Explained in detail in sections above.

Understanding series and progressions is critical for solving problems involving pattern recognition and prediction in mathematics, physics, finance, and other fields.
Progressions help to analyze data sequences, model behaviors, and predict future trends using established mathematical concepts, while series are useful in approximating functions and calculating areas or inventories.