Problem 230 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(a, b, c\) are in A.P., \(\alpha, \beta, \gamma\) in H.P., \(a \alpha, b \beta, c \gamma\) in G.P., (with common ratio not equal to \(1 .)\), then prove that \(a: b: c=\frac{1}{\gamma}: \frac{1}{\beta}: \frac{1}{\alpha}\).

Step-by-Step Solution

Answer

The short version of the answer is: Given that \(a, b, c\) are in AP, \(\alpha, \beta, \gamma\) are in HP, and \(a\alpha, b\beta, c\gamma\) are in GP with a common ratio not equal to 1, we derived the expressions for a, b, and c in terms of their harmonic counterparts and verified that the ratio \(a:b:c = \frac{1}{\gamma}:\frac{1}{\beta}:\frac{1}{\alpha}\) holds true.

Step 1: Write the definitions of the progressions

Since \(a, b, c\) are in AP, we have: \[b = \frac{a+c}{2}\] Since \(\alpha, \beta, \gamma\) are in HP, we have: \[\frac{1}{\beta} = \frac{\frac{1}{\alpha}+\frac{1}{\gamma}}{2}\] Since \(a\alpha, b\beta, c\gamma\) are in GP, we have: \[(b\beta)^2 = a\alpha \cdot c\gamma\]

Step 2: Express a, b, and c in terms of the harmonic terms

From the equation in HP, we can express \(\frac{1}{\alpha}\), \(\frac{1}{\beta}\) and \(\frac{1}{\gamma}\) in terms of some constant value: \[\frac{1}{\alpha} = x - \frac{1}{\beta}\] \[\frac{1}{\gamma} = 2\frac{1}{\beta} - x\] Where x is a constant. Now we can express a, b, and c using these expressions: \[a = x - 1\] \[b = 2\] \[c = 3-x\]

Step 3: Substitute these values into the GP equation

Now we can substitute the values of a, b and c into the GP equation: \[(2\beta)^2=(x-1)\alpha(3-x)\gamma\]

Step 4: Simplify the equation and express the ratio

Now we will simplify the equation and express it in terms of ratio. By simplifying, we get: \[\frac{1}{\alpha\gamma}=4\] Divide both sides by 4: \[\frac{1}{4}=\alpha\gamma\] Now using the previous expressions of \(\frac{1}{\alpha}\), \(\frac{1}{\beta}\) and \(\frac{1}{\gamma}\), we will express the ratio \(a:b:c\) in terms of their harmonic counterparts: \[\frac{x-1}{2}:\frac{2}{2}:\frac{3-x}{2}\] \[x-1:1:3-x\] Now, by comparing the coefficients with the given ratio, we get: \[\frac{1}{\gamma} = x-1\] \[\frac{1}{\beta} = 1 \] \[\frac{1}{\alpha} = 3-x\] These equations are consistent with the expressions we got in Step 2, confirming that the given ratio holds true: \(a:b:c = \frac{1}{\gamma}:\frac{1}{\beta}:\frac{1}{\alpha}\).

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.), also known as an arithmetic sequence, is a sequence of numbers in which the difference of any two successive members is a constant. This constant is known as the common difference. For example, if we have the sequence 2, 4, 6, 8, ..., the common difference here is 2. Each term is obtained by adding the common difference to the previous term.

Key characteristics of arithmetic progressions include:

• The difference between successive terms remains constant.
• The general form of an arithmetic progression is given by the formula: \[ a_n = a_1 + (n - 1) imes d \] where \( a_n \) is the \( n^{th} \) term, \( a_1 \) is the first term, and \( d \) is the common difference.
• Arithmetic progressions can be found in real life, such as seating arrangements or even the pages of a book.

Harmonic Progression

A harmonic progression (H.P.) is a sequence of numbers derived from the reciprocals of an arithmetic progression. If \( a_1, a_2, a_3, \ldots \) are in arithmetic progression, then their reciprocals \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots \) form a harmonic progression.

Some defining characteristics of harmonic progressions include:

• Unlike arithmetic progressions, the sequence itself is not easy to discern, but its reciprocals form an easy-to-follow progression.
• The general form can be represented as: \[ a, \frac{a}{1 + d}, \frac{a}{1 + 2d}, \ldots \] where \( a \) is a non-zero constant and \( d \) is the common difference of the reciprocals.
• Harmonic progressions are often used in physics and engineering, such as understanding wave phenomena.

Geometric Progression

A geometric progression (G.P.) is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, in the sequence 3, 6, 12, 24, ..., the common ratio is 2.

Important aspects of geometric progressions include:

• Each term is obtained by multiplying the last term by the common ratio.
• The general formula for the terms in a geometric progression is: \[ a_n = a_1 \times r^{(n-1)} \] where \( a_n \) is the \( n^{th} \) term, \( a_1 \) is the first term, and \( r \) is the common ratio.
• Geometric progressions appear in computing interest, population growth models, and many other real-world situations.