Problem 66 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(a_{1}, a_{2}, \ldots, a_{10}\) be a G.P. If \(\frac{a_{3}}{a_{1}}=25\), then \(\frac{a_{9}}{a_{5}}\) equals : [Jan. 11, 2019 (I)] (a) \(5^{4}\) (b) \(4\left(5^{2}\right)\) (c) \(5^{3}\) (d) \(2\left(5^{2}\right)\)

Step-by-Step Solution

Answer

The answer is (a) \( 5^4 \).

Step 1: Understanding the Geometric Progression (G.P.)

A geometric progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio, denoted as \( r \). Thus, the sequence is denoted as \( a, ar, ar^2, ar^3, \dots \).

Step 2: Identifying the Given Information

We are given that \( \frac{a_3}{a_1} = 25 \). In a G.P., \( a_3 = ar^2 \) and \( a_1 = a \). Therefore, \( \frac{ar^2}{a} = 25 \), which simplifies to \( r^2 = 25 \).

Step 3: Solving for the Common Ratio

Since \( r^2 = 25 \), we find \( r = 5 \) or \( r = -5 \). We'll consider the positive case \( r = 5 \) for simplicity, as either will lead to the same magnitude

Step 4: Express \( \frac{a_9}{a_5} \) in Terms of the Common Ratio

The terms \( a_9 \) and \( a_5 \) in a G.P. can be expressed as \( a_9 = ar^8 \) and \( a_5 = ar^4 \). Therefore, \( \frac{a_9}{a_5} = \frac{ar^8}{ar^4} = r^4 \).

Step 5: Calculating \( r^4 \) Using \( r = 5 \)

Substitute \( r = 5 \) into the expression \( r^4 \). Therefore, \( r^4 = 5^4 = 625 \).

Step 6: Final Step: Choosing the Correct Answer

Given that \( \frac{a_9}{a_5} = 5^4 = 625 \), the answer to the question is the option that equals \( 5^4 \), which is option (a) \( 5^4 \).

Key Concepts

Understanding the Common Ratio in Geometric Progression

When discussing a geometric progression, one crucial concept is the common ratio. This is the factor by which we scale each term to reach the next in the sequence. For instance, if your first number is 2 and your common ratio is 3, the sequence progresses as 2, 6, 18, and so on.
The common ratio is derived from dividing any term in the sequence by its preceding term. If you’re given three terms of a geometric progression like in our exercise, it's helpful to remember this formula:

• First term, often denoted as \( a \)
• Second term is \( ar \)
• Third term is \( ar^2 \)

It means if you have \( \frac{a_3}{a_1} = 25 \), we express it as \( \frac{ar^2}{a} = 25 \). Simplifying gives \( r^2 = 25 \), leading to \( r = 5 \) or \( r = -5 \). This constant "r" is what dictates the shade of progression in your sequence.

Making Sense of Sequence Terms

A sequence in mathematics is essentially a list of numbers in a specific order. In a geometric sequence, you start with a base value and subsequently obtain other values by multiplying the previous number by the common ratio.
For example, let's say you start with a base number of 3 and use a common ratio of 2. Your sequence would look like 3, 6, 12, 24, and so on.
Each value is a term in the sequence and is denoted systematically:

• \( a_1 \) is the first term, \( a \)
• \( a_2 \) is \( ar \)
• \( a_3 \) is \( ar^2 \)

To find \( \frac{a_9}{a_5} \), you identify \( a_9 \) as \( ar^8 \) and \( a_5 \) as \( ar^4 \). Dividing these gives us \( r^4 \), making calculations straightforward once we know the value of \( r \). This systematic approach aids in finding terms and positioning within any geometric progression.

Exploring Exponential Growth in Geometric Progressions

Geometric progressions are powerful because they demonstrate exponential growth, a rapid increase due to each term's multiplication by a constant factor. This is what makes sequences like \( a, ar, ar^2, ar^3, \ldots \) vastly increase as you progress further down the sequence.
Consider a microbial population growing under ideal conditions, represented by a geometric sequence with a common ratio larger than 1. Starting with one bacterium, if the population doubles daily, by the tenth day, there would be \( a_{10} = ar^9 \) bacteria, substantially more than the initial count.
This concept reflects real-world situations like compound interest, population models, and any scenario illustrating multiplicative growth. It's important to note: as the common ratio increases in a geometric sequence, so does the "speed" or rate of exponential growth. Thus, understanding both the sequence's structure and the implications of the ratio is vital for recognizing patterns and predicting future values.