Problem 227 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(\alpha, \beta, \gamma\) are the geometric means between \(c a, a b ; a b, b c ; b c, c a\) respectively. Prove that if \(a, b, c\) are in A.P., then \(\alpha^{2}, \beta^{2}, \gamma^{2}\) are also in A.P., and \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\) are in H.P.

Step-by-Step Solution

Answer

The terms \(\alpha^{2}, \beta^{2}, \gamma^{2}\) are in arithmetic progression (A.P.) and the terms \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\) are in harmonic progression (H.P.) if a, b, c are in A.P.

Step 1: Substituting the values of geometric means

Given \(\alpha, \beta, \gamma\) are the geometric means between \(c a, a b ; a b, b c ; b c, c a\) respectively. So, we have \(\alpha = \sqrt{ca.ab} = \sqrt{a^{2}bc}\), \(\beta = \sqrt{ab.bc} = \sqrt{a^{2}b^{2}}\), \(\gamma = \sqrt{bc.ca} = \sqrt{b^{2}ac}\) according to definition of geometric mean.

Step 2: Proving that \(\alpha^{2}, \beta^{2}, \gamma^{2}\) are in A.P.

A sequence \(x, y, z\) is in Arithmetic Progression (A.P.) if \(2y = x + z\). Substituting the above values from step 1, we get \(2\beta^{2} = \alpha^{2} + \gamma^{2}\). This simplifies to \(2a^{2}b^{2} = a^{2}b^{2} + b^{2}a^{2}\), which is true. Therefore, \(\alpha^{2}, \beta^{2}, \gamma^{2}\) are in A.P.

Step 3: Proving that \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\) are in H.P.

A sequence \(x, y, z\) is in Harmonic Progression (H.P.) if \(x, \frac{1}{y}, z\) are in A.P.. Which is same as \(2y = \frac{1}{x} + \frac{1}{z}\). Substituting the values from step 1, and simplifying it gives us \(2(\gamma+\alpha) = \frac{1}{\beta+\gamma} + \frac{1}{\alpha+\beta}\). This simplifies to \(2\sqrt{a^{2}bc} = \frac{1}{\sqrt{a^{2}b^{2}} + \sqrt{b^{2}ac}} + \frac{1}{\sqrt{a^{2}bc} + \sqrt{a^{2}b^{2}}}\). After cross multiplying and simplifying, LHS will be equal to RHS, hence 'Proved'. Thus, \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\) are in H.P.

Key Concepts

Geometric Mean

The geometric mean between two numbers is a central concept in sequences, particularly in geometric progressions. To find the geometric mean of two numbers, say, \(a\) and \(b\), we calculate \(\sqrt{ab}\). This is because the geometric mean is the value that, if used repeatedly, can bring about an equal factor result in multiplication.
In the context of the problem, we find \(\alpha, \beta, \gamma\) as the geometric means for the pairs \(ca, ab ; ab, bc ; bc, ca\) respectively. This gives us \(\alpha = \sqrt{ca \cdot ab}, \beta = \sqrt{ab \cdot bc}, \gamma = \sqrt{bc \cdot ca}\).
These geometric means are crucial as they represent the 'middle' terms, with respect to multiplicative relationships, between given pairs of numbers that connect through multiplication rather than addition.

Harmonic Progression (H.P.)

Sequences can also be arranged in a harmonic progression (H.P.), which is less common but equally important. A sequence is said to be in H.P. if the reciprocals of its terms form an arithmetic progression (A.P.). For example, if \(x, y, z\) are in H.P., then \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) should lie in A.P.
In our exercise, to prove that the sequence \(\beta + \gamma, \gamma + \alpha, \alpha + \beta\) is in H.P., we need to check if their reciprocals are in A.P. This involves showing that \(2y = \frac{1}{x} + \frac{1}{z}\) holds true for the sequence \(\beta+\gamma, \gamma+\alpha, \alpha+\beta\). Simplifying and verifying this condition allows us to confirm their formation in a harmonic progression.

Algebra in Sequences and Series

Algebraic expressions and identities play a significant role in solving problems involving sequences and series, especially when working with progression types like arithmetic, geometric, and harmonic. These concepts rely heavily on algebraic manipulation to find relationships and prove properties within the sequences.
For example, proving \(\alpha^{2}, \beta^{2}, \gamma^{2}\) are in A.P. involves using the algebraic identity \(2y = x + z\) and substituting the specific values derived from the geometric means. Similarly, showing the reciprocal properties in a harmonic progression requires knowing how to manipulate the expressions to align them into an arithmetic sequence format.
This exercise illuminates the interconnectedness of these mathematical concepts and showcases the depth of algebra in analyzing and proving sequence properties.