Problem 106 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the first and the \((2 n-1)\) th terms of an A.P., G.P. and H.P. are equal and their \(n\)th terms are \(a, b, c\) respectively, then (A) \(a=b=c\) (B) \(a \geq b \geq c\) (C) \(a+c=b\) (D) \(a c-b^{2}=0\)

Step-by-Step Solution

Answer

(D) \(ac-b^2 = 0\).

Step 1: Understand the Problem

We need to find the relationship between the functions of Arithmetic Progression (A.P.), Geometric Progression (G.P.), and Harmonic Progression (H.P.) when their first and (2n-1)th terms are equal. Given the nth terms are denoted by a, b, and c, we need to verify which option (A, B, C, D) holds true.

Step 2: Define the A.P., G.P., and H.P. Terms

For an A.P., the nth term is given by \( a_n = a_1 + (n-1) d \). For a G.P., the nth term is \( b_n = b_1 r^{n-1} \). For an H.P., the nth term is \( c_n = \frac{1}{f_1 + (n-1) j} \). Here, we need to find expressions for the first term (first term) and the (2n-1)th terms of these progressions.

Step 3: Set Equal Conditions for the First Terms

First, we know that the first term in each sequence is equal, \( a_1 = b_1 = c_1 \), thus: For an A.P.: \( a_1 = A \)For a G.P.: \( b_1 = A \)For an H.P. (inverted A.P.): \( c_1 = \frac{1}{A} \).

Step 4: Set Equal Conditions for the (2n-1)th Terms

Similarly, the (2n-1)th terms are equal, \( a_{2n-1} = b_{2n-1} = c_{2n-1} \): A.P.: \( a_{2n-1} = A + (2n-2) d = A \)G.P.: \( b_{2n-1} = A r^{2n-2} = A \)H.P.: \( c_{2n-1} = \frac{1}{A + (2n-2) j} = A \).

Step 5: Derive nth Terms Based on Conditions

Now, derive expressions for the nth terms:A.P.: \( a = A + (n-1) d \)G.P.: \( b = A r^{n-1} \)H.P.: \( c = \frac{1}{A + (n-1) j} \).

Step 6: Analyse the Relations Between a, b, and c

Now we check the validity of options provided:- Since A.P., G.P., and H.P. have the same first and (2n-1)th terms, use the relation; \(a=c=b\).- Option D implies \(ac-b^2=0\), this is equivalent to \( a=c=b \).

Step 7: Confirm the Correct Option

Realize that if \(a=c=b\), \(ac-b^2=0\), thus option (D) is demonstrated as true under these conditions.

Key Concepts

Geometric Progression

A geometric progression (G.P.) is a sequence of numbers where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the common ratio. This kind of sequence can be represented as:

• First term: \( T_1 = a \)
• Second term: \( T_2 = a \cdot r \)
• Third term: \( T_3 = a \cdot r^2 \)

Where \( r \) is the common ratio. If \( r > 1 \), the sequence grows exponentially, while if \( 0 < r < 1 \), it decreases towards zero.
G.P.s are particularly interesting since they appear naturally in various growth models, including population growth and financial interest calculations. The formula for the nth term of a geometric progression is given by \( T_n = a \cdot r^{n-1} \). In exercises similar to the one provided, solving for equal terms involves setting up equations to find the common ratio such that the conditions hold true.

Harmonic Progression

A harmonic progression (H.P.) is a sequence of numbers derived from the reciprocals of an arithmetic progression (A.P.). Suppose you're given an arithmetic progression with terms \( a_1, a_2, a_3, \ldots \), the corresponding harmonic progression would have terms \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots \).
The formula for the nth term in a harmonic progression can be expressed as the reciprocal of the corresponding A.P. term:

• If \( a_n = a_1 + (n-1) \cdot d \), then \( H_n = \frac{1}{a_n} \)

Harmonic progressions are significant in physics and engineering, particularly in the study of systems characterized by wave and frequency behaviors. In the exercise context, finding that the first and \((2n-1)\)th terms are equal requires manipulating these reciprocal relationships, as reflected in their derived equations.

Progression Comparisons

Comparing different types of progressions, like arithmetic, geometric, and harmonic, often involves analyzing how each sequence behaves under various conditions. In the provided exercise, the primary task is to scrutinize the relationships between these progressions.

• Arithmetic Progression (A.P.): Deals with linear changes which involve a constant addition (or subtraction), represented by its common difference.
• Geometric Progression (G.P.): Incorporates exponential changes through a multiplying factor, known as the common ratio.
• Harmonic Progression (H.P.): Involves inverse relationships, derived from the reciprocals of an arithmetic sequence.

The beauty of analyzing progressions lies in understanding how uniquely each affects the outcome of sequences, particularly when conditions such as having equal first and \((2n-1)\)th terms arise. For the equation \( ac - b^2 = 0 \), rooted in this comparison, understanding the equivalence between terms becomes critical in determining which sequences meet these criteria. Consequently, these comparisons are not merely academic exercises, but real-world tools for interpreting trends and behaviors.