Problem 89 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the \((m+1)\) th, \((n+1)\) th and \((r+1)\) th terms of an A.P. are in G.P. and \(m, n, r\) are in H.P., then the ratio of the first term of the A.P. to its common difference is (A) \(\frac{n}{3}\) (B) \(-\frac{n}{3}\) (C) \(\frac{n}{2}\) (D) \(-\frac{n}{2}\)

Step-by-Step Solution

Answer

The ratio of the first term of the A.P. to the common difference is \(-\frac{n}{3}\).

Step 1: Identify A.P. Terms

In an arithmetic progression (A.P.), the \((m+1)\)th term is given by \(a + md\), the \((n+1)\)th term is \(a + nd\), and the \((r+1)\)th term is \(a + rd\), where \(a\) is the first term and \(d\) is the common difference.

Step 2: Express G.P. Condition

The terms \(a + md\), \(a + nd\), and \(a + rd\) are in geometric progression (G.P.). This means the ratio of consecutive terms is constant, so we have \((a+nd)^2 = (a+md)(a+rd)\).

Step 3: Express H.P. Condition

The numbers \(m, n, r\) are in harmonic progression (H.P.), meaning the reciprocals \(\frac{1}{m}, \frac{1}{n}, \frac{1}{r}\) are in arithmetic progression. This gives the equation \(\frac{2}{n} = \frac{1}{m} + \frac{1}{r}\).

Step 4: Simplify G.P. Equation

Simplifying \((a + nd)^2 = (a + md)(a + rd)\), we expand terms to get \(a^2 + 2and + n^2d^2 = a^2 + ard + amd + mrd^2\). Cancel out \(a^2\) and collect like terms.

Step 5: Solve for Common Terms

From the simplification, obtain \(2and - amd - ard = mrd^2 - n^2d^2\). Factor this equation as \((2n - m - r)a = (mr - n^2)d\).

Step 6: Use H.P. Relation

Substitute \(2n = m + r\) from the H.P. condition into the simplified equation to simplify further: \((2n - (m + r))a = (mr - n^2)d\). This simplifies to \(0 = (mr - n^2)d\), which implies \(mr = n^2\).

Step 7: Divide and Resolve

Now, from the relationship \((2n - m - r)a = (mr - n^2)d\), and knowing \(mr = n^2\), conclude that \((mr - n^2) = 0\) implies that \(a = -\frac{nd}{3}\), simplifying gives \(\frac{a}{d} = -\frac{n}{3}\).

Step 8: Select the Correct Ratio

The calculated ratio of the first term to the common difference is \(-\frac{n}{3}\). Therefore, the answer is option (B).

Key Concepts

Geometric Progression

A Geometric Progression (G.P.) 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. In the context of the provided problem, the terms \(a + md\), \(a + nd\), and \(a + rd\) form a G.P. if there exists a common ratio, \(r\), such that the ratio of the second term to the first is equal to the ratio of the third term to the second.
This condition is expressed by the equation:

• \((a+nd)^2 = (a+md)(a+rd)\)

Understanding this equation will help you realize how G.P. enforces a strict multiplicative symmetry across terms, such that each term is a multiple of its predecessor by the common ratio. This property is essential when linking it with A.P. terms to form a solvable algebraic equation. The intrinsic structure of G.P. allows us to connect terms of different progressions in a meaningful mathematical relationship.

Harmonic Progression

In a Harmonic Progression (H.P.), the reciprocals of the terms follow an Arithmetic Progression (A.P.). In practical terms, this means if \(m, n, r\) are in H.P., then \(\frac{1}{m}, \frac{1}{n}, \frac{1}{r}\) are in an A.P., establishing a linear relation between them. Specifically, it can be presented in the form:

• \(\frac{2}{n} = \frac{1}{m} + \frac{1}{r}\)

In the problem, this relationship helps derive an important step that simplifies the solution process. It reveals the condition that links \(m, n,\) and \(r\), specifically yielding \(2n = m + r\). This derivation is crucial as it helps relate the structure of H.P. to the terms of A.P., providing a stepping stone to deducing the requirements for G.P. and solving for the desired ratio of A.P. terms.

Common Difference

The Common Difference in an Arithmetic Progression (A.P.) is the consistent interval difference between consecutive terms. In the solution, the first term \(a\) and the common difference \(d\) form the core of understanding the progression's pattern: each term is precisely \(d\) units more than the one preceding it.
When attempting to find the ratio \(\frac{a}{d}\), it is essential to draw from relationships established by other progressions (like G.P. and H.P.). The A.P. definition establishes that:

• \(a + nd\) is a specific term in the progression, just as \(a\), \(a + md\), and \(a + rd\) are.

The interplay of Common Difference with the conditions of G.P. and H.P. allows the chosen portion of A.P. terms to settle into a formula that we can manipulate using algebraic methods. Solving the equation \((2n - m - r)a = (mr - n^2)d\) and using the H.P. relation \(2n = m + r\) simplifies the equation enough to reveal the sought-after ratio, leading to \(\frac{a}{d} = -\frac{n}{3}\). This result highlights the vital role of understanding not just individual terms, but how they mesh through their common difference to align with alternate progression patterns.