Problem 62 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(a, b\) and \(c\) be in G.P. with common ratio \(r\), where \(a \neq 0\) and \(0

Step-by-Step Solution

Step 1: Define Geometric Progression

We know that in a geometric progression (G.P.), consecutive terms are related by a common ratio, \( r \). This means:1. \( b = ar \)2. \( c = ar^2 \)

Step 2: Define Arithmetic Progression

In an arithmetic progression (A.P.), the difference between consecutive terms remains constant:1. Let the A.P. be defined as \( x_1, x_2, x_3, \ldots \)2. Given terms of A.P. are \( x_1 = 3a \), \( x_2 = 7b \), and \( x_3 = 15c \).3. The common difference \( d \) is defined as \( d = x_2 - x_1 = x_3 - x_2 \).

Key Concepts

Geometric Progression

A Geometric Progression (G.P.) is a sequence where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number known as the common ratio.
If you have terms like \( a, b, \) and \( c \) in a geometric sequence, you can express these terms using the formula:

• The second term \( b \) can be expressed as \( ar \), where \( a \) is the first term and \( r \) is the common ratio.
• The third term \( c \) can be expressed as \( ar^2 \).

This sequence keeps building upon the previous term by multiplying it by \( r \).

G.P. is incredibly useful in understanding various phenomena, such as exponential growth or decay, in mathematical contexts and even in real-world scenarios. For example, the population growth often follows a geometric progression.

common ratio

The common ratio is a crucial concept in geometric progressions. It defines the constant factor between consecutive terms in the sequence. When you move from one term to the next in a geometric progression, you multiply by this common ratio to find the subsequent term.

Let's examine the common ratio a bit more:

• If your G.P. has terms \( a, b, \) and \( c \), the common ratio \( r \) is found by dividing any term by its previous term, such as \( r = \frac{b}{a} = \frac{c}{b} \).
• It's crucial to note that the common ratio can be less than 1, which indicates a reducing sequence, or greater than 1, leading to an increasing sequence.
• The conditions given in the exercise are noteworthy: \( 0 < r \leq \frac{1}{2} \), suggesting the sequence is shrinking or stable but not increasing.

Understanding the common ratio enables you to predict any term in the sequence given the initial term and the number of terms you want to find.

common difference

In contrast to geometric progression, an Arithmetic Progression (A.P.) involves a sequence of numbers where the difference between any two successive terms is constant, known as the common difference.

To see how this works, consider the sequence formed by terms like \( 3a, 7b, \) and \( 15c \). To find the common difference \( d \) between these terms:

• Calculate \( d = x_2 - x_1 \), substituting \( x_1 = 3a \) and \( x_2 = 7b \).
• Also, \( d = x_3 - x_2 \), and substituting \( x_2 = 7b \) and \( x_3 = 15c \).

This common difference remains uniform throughout the progression, making it possible to determine future terms in the sequence straightforwardly by adding \( d \) to known terms.

An arithmetic progression is found in regular patterns such as calendar days, equal payments in an installment plan, etc., where increments remain steady over time.