Problem 102 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Does there exist a geometric progression containing 27,8 and 12 as three of its terms? If it exists, how many such progressions are possible?

Step-by-Step Solution

Answer

There does not exist a geometric progression containing the numbers 27, 8, and 12 as its terms because no common ratio exists that satisfies the given conditions.

Step 1: Checking if a geometric progression can exist

For a geometric progression to exist, the ratio between any two consecutive terms must be the same. Let's try to determine if a common ratio exists between the given numbers. Let \(a\) be the first term, \(r\) be the common ratio, and \(n\) represent the position of the terms. Given the terms 27, 8, and 12, we can write three equations: 1. \(a * r^{n_1} = 27\) 2. \(a * r^{n_2} = 8\) 3. \(a * r^{n_3} = 12\) We need to determine if there's a common ratio and progression that satisfy all three equations.

Step 2: Finding the possible common ratios

Let's divide the equations to find the possible ratios: \(\frac{27}{8} = (\frac{a * r^{n_1}}{a * r^{n_2}}) = r^{n_1 - n_2}\) \(\frac{12}{8} = (\frac{a * r^{n_3}}{a * r^{n_2}}) = r^{n_3 - n_2}\) \(\frac{27}{12} = (\frac{a * r^{n_1}}{a * r^{n_3}}) = r^{n_1 - n_3}\) Now, we need to check if there exist integer powers that satisfy these relationships: \(\frac{27}{8} = r^{n_1 - n_2}\) \(\frac{12}{8} = r^{n_3 - n_2}\) \(\frac{27}{12} = r^{n_1 - n_3}\)

Step 3: Checking for integer powers

If we simplify the fractions, we get: \(\frac{27}{8} = r^{n_1 - n_2}\) \(\frac{3}{2} = r^{n_3 - n_2}\) \(\frac{9}{4} = r^{n_1 - n_3}\) We can see that the second equation, \(\frac{3}{2} = r^{n_3 - n_2}\), implies that there should be a common ratio \(r\) that when raised to an integer power (difference of positions) is equal to \(\frac{3}{2}\). Since all our ratios are fractions, there's no integer value for the exponents that satisfies this condition.

Step 4: Concluding if the geometric progression exists

As we have determined that no common ratio exists that satisfies the given conditions, we can conclude that there does not exist a geometric progression containing the numbers 27, 8, and 12 as its terms.

Key Concepts

Common Ratio

Understanding the concept of a common ratio is essential when studying geometric progressions. A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio (r). This ratio is a cornerstone of geometric sequences because it provides the means to quickly identify any term in the sequence.

For example, in a sequence where each number is double the previous one, the common ratio would be 2. This is because each term is obtained by multiplying the previous one by 2. In mathematical terms, for a GP with the first term (a), the nth term (an) is given by:
\( an = a \times r^{(n-1)} \).

The exercise involving the numbers 27, 8, and 12 seeks to identify if a geometric progression could exist among them by finding a consistent common ratio. However, as demonstrated in the solution, it was impossible to find such a ratio that satisfied the relationships between these particular numbers. Thus, the key takeaway is that an underlying common ratio must be consistent across all terms for a GP to be valid.

Sequence of Numbers

A sequence of numbers is a set of numbers that follow a particular rule which determines the relationship between successive numbers. In mathematics, there are various types of sequences, such as arithmetic, geometric, and Fibonacci, to name a few. The defining characteristic of a sequence is that we can predict subsequent numbers based on previous ones using a specific formula.

For a geometric progression, the rule is that we multiply a term by the common ratio to get the next term. This multiplication factor remains constant throughout the entire sequence. It's crucial to note that the concept of sequence positions, denoted by 'n', plays a significant role. Positions help to identify a particular term within the sequence and are utilized to calculate the value of that term based on the given formula.

In the context of our exercise, we see that even though 27, 8, and 12 are numbers in sequence, they don't create a geometric sequence because there is no common ratio that can maintain the progression from one term to the next. This property of sequences helps to classify a set of numbers as part of a specific type of progression or to rule out their inclusion as such.

Exponential Equations

Exponential equations are equations where the variable appears in the exponent. In the domain of sequences and progressions, such equations are frequently encountered, particularly when dealing with geometric progressions.

These equations take the form \( a \cdot r^n = b \), where 'a' is a constant, 'r' is the base of the exponent, 'n' is the exponent, and 'b' is the result. Solving exponential equations often requires finding the value of the variable in the exponent that makes the equation true. This can involve logarithmic functions or, in the case of sequences, simply deducing the pattern or the relationship between the terms.

In our textbook example, various exponential equations were crafted to establish potential common ratios. The comparison of these exponential equations aimed to confirm the possibility of integer exponents that would validate the GP. However, the outcome revealed that no integer solutions existed for the given values, therefore concluding that a geometric progression with the terms 27, 8, and 12 can't exist. Such problems underscore the importance of understanding exponential equations when studying geometric progressions.