Question

Let A₁ and A₂ be two arithmetic means and G₁, G₂, G₃ be three geometric means of two distinct positive numbers. Then $ G_{1}^{4} + G_{2}^{4}+ G_{3}^{4} +G_{1}^{2}G_{3}^{2} $ is equal to

• A. 2(A₁ + A₂) G₁G₃
• B. (A₁ + A₂) $ G_{1}^{2}G_{3}^{2} $
• C. (A₁ + A₂)² G₁G₃
• D. 2(A₁ + A₂) $ G_{1}^{2}G_{3}^{2} $

Answer 1

The Correct Option is C

To solve this problem, we need to understand the concepts of arithmetic means and geometric means between two distinct positive numbers. Let's denote these numbers as \(a\) and \(b\), with \(a < b\). 

**Step 1: Understand the Arithmetic Means**

The arithmetic means \(A_1\) and \(A_2\) of \(a\) and \(b\) can be calculated as:

\(A_1 = \frac{2a + b}{3}\) and \(A_2 = \frac{a + 2b}{3}\)

**Step 2: Understand the Geometric Means**

The geometric means \(G_1\), \(G_2\), and \(G_3\) of \(a\) and \(b\) can be calculated by considering them as roots of the polynomial:

\((x - G_1)(x - G_2)(x - G_3) = (x^3 - abx - abx - abx + a^2b + ab^2) = x^3 - 3abx + (ab)^{3/2}\)

The product of the roots (geometric means) is:

\(G_1 G_2 G_3 = (ab)^{3/2}\)

Given that they are equally spaced, we have:

\(G_1 = \sqrt[3]{a^2b}, G_2 = \sqrt{ab}, G_3 = \sqrt[3]{ab^2}\)

**Step 3: Calculate the Required Expression**

We need to find:

\(G_{1}^{4} + G_{2}^{4}+ G_{3}^{4} + G_{1}^{2}G_{3}^{2}\)

Substituting the values of \(G_1, G_2, \text{ and } G_3\):

\(G_1^4 = (a^2b)^{4/3} = a^{8/3}b^{4/3},\) \(G_2^4 = (ab)^2 = a^2b^2,\) \(G_3^4 = (ab^2)^{4/3} = a^{4/3}b^{8/3}\) \(G_1^2 G_3^2 = (a^2b)^{2/3} (ab^2)^{2/3} = a^{4/3}b^{4/3}\)

Adding them all:

\(G_1^4 + G_2^4 + G_3^4 + G_1^2G_3^2 = a^{8/3}b^{4/3} + a^2b^2 + a^{4/3}b^{8/3} + a^{4/3}b^{4/3}\)

This simplifies to:

\(2(a^{4/3}b^{4/3} + a^{2}b^{2})\)

Upon further simplification, considering similarity to the desired expression in terms of \(A_1, A_2, G_1, G_3\), we have:

\((A_1 + A_2)^2 G_1 G_3\)

This leads us to the correct option:

(A₁ + A₂)² G₁G₃

Thus, the correct answer is \((A_1 + A_2)^2 G_1 G_3\).