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

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 is \(-\frac{n}{3}\), hence option (B).

Step 1: Understand the Problem Statement

We have an arithmetic progression (A.P.) with terms defined by their indices \((m+1), (n+1), (r+1)\). It is stated that these terms form a geometric progression (G.P.). Also, the indices \(m, n, r\) are in harmonic progression (H.P.). We need to find the ratio of the first term of the A.P. to its common difference.

Step 2: Express Terms in A.P.

In an A.P., the \((k+1)\)th term is given by \(a_k = a + kd\), where \(a\) is the first term and \(d\) is the common difference. Therefore,- \((m+1)\)th term: \(a_{m+1} = a + md\)- \((n+1)\)th term: \(a_{n+1} = a + nd\)- \((r+1)\)th term: \(a_{r+1} = a + rd\)

Step 3: Set Up the Condition for G.P.

In a G.P., the square of the middle term is equal to the product of the other two terms. Therefore, for the terms \(a_{m+1}, a_{n+1}, a_{r+1}\) in G.P., we have:\[(a + nd)^2 = (a + md)(a + rd)\]

Step 4: Set Up Condition for H.P.

For three numbers \(m, n, r\) in H.P., their reciprocals form an arithmetic progression. Thus:\[2\left(\frac{1}{n}\right) = \frac{1}{m} + \frac{1}{r}\]Which simplifies to:\[2nr = mr + mn\]

Step 5: Solve Both Equations

Substituting the H.P. condition into the equation derived from the G.P.:1. From \((a + nd)^2 = (a + md)(a + rd)\), expand both sides:\[a^2 + 2and + n^2d^2 = a^2 + ard + amd + mr d^2\]2. Rearrange:\[2and + n^2d^2 = ard + amd + mrd^2\]3. Using \(2mr = mr + mn\), we deduce that:\(n = \frac{2mr}{m + r}\).4. Simplifying gives the ratio \(\frac{a}{d} = -\frac{n}{3}\).

Step 6: Check Available Options

From the calculated ratio \(\frac{a}{d} = -\frac{n}{3}\), compare this with the provided options:- (A) \(\frac{n}{3}\)- (B) \(-\frac{n}{3}\)- (C) \(\frac{n}{2}\)- (D) \(-\frac{n}{2}\)Hence, option (B) is the correct answer.

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. This concept is an essential part of mathematics since it represents exponential growth or decay.

• For example, in the sequence 2, 4, 8, 16, the common ratio is 2, since each term is multiplied by 2 to get the next.
• If the common ratio is greater than one, the terms increase; if it is between zero and one, the terms decrease.

In the context of the problem, the \((m+1)\)th, \((n+1)\)th, and \((r+1)\)th terms of an arithmetic progression are said to be in a geometric progression. This means:\[(a_{n+1})^2 = a_{m+1} \cdot a_{r+1}\]Understanding how to solve for such conditions in progressions improves problem-solving skills, especially in series and sequences.

Harmonic Progression

A Harmonic Progression (H.P.) is a sequence of numbers derived from the reciprocals of an arithmetic progression (A.P.). In other words, if a sequence \(a, b, c, \ldots\) is in H.P., then the reciprocals \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \ldots\) form an A.P.

• H.P. is useful to show relationships between different mathematical phenomena through the harmonic mean.
• For three terms \(m, n, r\) to be in H.P., their reciprocals must be in A.P.

In the given exercise, the indices \(m, n, r\) are in H.P., which translates to the condition:\[2 \left( \frac{1}{n} \right) = \frac{1}{m} + \frac{1}{r}\]This simplifies to establishing the relationship among \(m, n, r\) in terms of harmonic series, which aids in understanding their behavior in the sequence.

Common Difference

The Common Difference in an arithmetic progression (A.P.) is the constant amount by which each term in the sequence increases to get to the next term. It is a vital part of finding term positions in A.P.

• For an A.P., the nth term can be expressed as \(a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference.
• Various sequences, like linear growth patterns, rely on A.P. due to this constant addition pattern.

In this specific problem, determining the common difference helps solve the equation involving terms in both G.P and their relatedness in H.P. The task requires understanding its role to express terms in complex arrangements like those found in combinations of progressions, thus simplifying the problem to determine the correct ratio \(\frac{a}{d} = -\frac{n}{3}\).