Problem 74 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the \(2^{\text {nd }}, 5^{\text {th }}\) and \(9^{\text {th }}\) terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is : (a) 1 (b) \(\frac{7}{4}\) (c) \(\frac{8}{5}\) (d) \(\frac{4}{3}\)

Step-by-Step Solution

Answer

The common ratio is \(\frac{4}{3}\) (option d).

Step 1: Setting Up the Arithmetic Progression (A.P.)

To solve this problem, we first need to define the terms of the Arithmetic Progression (A.P.). Let's assume the first term of the A.P. is \(a\) and the common difference is \(d\). Therefore, the \(n\)-th term of an A.P. is given by \(a_n = a + (n-1)d\).

Step 2: Identifying the Terms in the A.P.

Since we know that the 2nd, 5th, and 9th terms of the A.P. form a Geometric Progression (G.P.), these terms can be written as:- 2nd term: \(a + d\)- 5th term: \(a + 4d\)- 9th term: \(a + 8d\)

Step 3: Setting Up the Geometric Progression (G.P.)

The terms \(a + d, a + 4d, a + 8d\) form a Geometric Progression (G.P.). In a G.P., the ratio between consecutive terms is constant. So, the common ratio \(r\) is given by the equation:\[r = \frac{a + 4d}{a + d} = \frac{a + 8d}{a + 4d}\].

Step 4: Solving for the Common Ratio

We now have two equations:1. \(\frac{a + 4d}{a + d} = r\)2. \(\frac{a + 8d}{a + 4d} = r\)Setting them equal:\[\frac{a + 4d}{a + d} = \frac{a + 8d}{a + 4d}\]Cross-multiply to solve for \(r\):\[(a + 4d)^2 = (a + d)(a + 8d)\].

Step 5: Expanding and Simplifying the Equation

Expand and simplify the equation:- Left hand side: \((a + 4d)^2 = a^2 + 8ad + 16d^2\)- Right hand side: \((a + d)(a + 8d) = a^2 + 8ad + ad + 8d^2 = a^2 + 9ad + 8d^2\)These simplify to:\[a^2 + 8ad + 16d^2 = a^2 + 9ad + 8d^2\].

Step 6: Finding the Correct Ratio

Cancel \(a^2\) from both sides:\[8ad + 16d^2 = 9ad + 8d^2\]Subtract \(8ad + 8d^2\) from both sides:\[8d^2 = ad\]This simplifies to \(a = 8d\).Since \(a = 8d\), substitute into the equation for \(r\):\[r = \frac{a + 4d}{a + d} = \frac{8d + 4d}{8d + d} = \frac{12d}{9d} = \frac{4}{3}\].

Step 7: Conclusion

The common ratio of the given G.P. is \(\frac{4}{3}\), which corresponds to option (d).

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 type of progression is easy to identify given:

• The sequence progresses multiplicatively rather than additively.
• Each term can be calculated using the formula for the nth term: \(a_n = a \cdot r^{(n-1)}\), where \(a\) is the first term and \(r\) is the common ratio.

In the world of sequences, understanding G.P. is key, especially when connecting it with other forms of sequences, such as Arithmetic Progression. It’s interesting to see how terms of an Arithmetic Progression can sometimes be arranged to form a Geometric Progression, as seen in cases like this exercise.

Common Ratio

The common ratio in a Geometric Progression is the constant factor between consecutive terms. It's a crucial value because it defines the nature and direction of the progression. For instance:

• If the common ratio is greater than 1, the terms will increase exponentially.
• If it is between 0 and 1, they'll decrease.
• A negative common ratio results in an alternating sequence of signs.

To find the common ratio in a G.P., you can use the formula \( r = \frac{a_{n+1}}{a_n} \), where \( a_{n+1} \) and \( a_n \) are consecutive terms. In problems involving Arithmetic Progressions forming a G.P., finding this common ratio typically involves setting equations against each other and solving algebraically, just like the exercise solution where equation manipulation was used to determine the ratio to be \( \frac{4}{3} \).

Sequences and Series

Sequences and series are fundamental concepts in mathematics, especially in understanding patterns among numbers. A sequence is simply a list of numbers arranged in a specific order, while a series is the sum of the terms of a sequence. There are many types of sequences and series, including:

• Arithmetic Sequences: where the difference between consecutive terms is constant (common difference).
• Geometric Sequences: where each term is multiplied by a constant (common ratio) to get the next.
• Harmonic and Fibonacci sequences, which have unique properties and patterns.

The study of sequences and series not only finds applications in solving academic exercises but also in various real-world contexts like finance and computer science. Mastering these concepts involves recognizing the type of sequence, understanding its properties, and using appropriate formulas and methods to solve relevant problems. This understanding enables learners to tackle complex mathematical challenges effectively, combining creativity with logical reasoning.