If `(a_2 a_3)/(a_1 a_4) = (a_2 + a_3)/(a_1 + a_4) = 3 ((a_2 - a_3)/(a_1 - a_4))` then `a_1, a_2, a_3, a_4` are in

Options

A. AP
B. GP
C. HP
D. none of these

Answer

To solve the problem, we start with the given equation:

\[ \frac{a_2 a_3}{a_1 a_4} = \frac{a_2 + a_3}{a_1 + a_4} = 3 \cdot \frac{a_2 - a_3}{a_1 - a_4} \]

We will analyze the first two parts of the equation:

1. **Step 1: Set up the first equality.**

From the first part, we have:
\[ \frac{a_2 a_3}{a_1 a_4} = \frac{a_2 + a_3}{a_1 + a_4} \]

Cross-multiplying gives:
\[ a_2 a_3 (a_1 + a_4) = (a_2 + a_3) a_1 a_4 \]

Expanding both sides:
\[ a_2 a_3 a_1 + a_2 a_3 a_4 = a_1 a_4 a_2 + a_1 a_4 a_3 \]

2. **Step 2: Rearranging the equation.**

Rearranging the equation gives:
\[ a_2 a_3 a_1 + a_2 a_3 a_4 - a_1 a_4 a_2 - a_1 a_4 a_3 = 0 \]

Grouping the terms:
\[ a_2 (a_3 a_1 - a_1 a_4) + a_3 (a_2 a_4 - a_1 a_4) = 0 \]

3. **Step 3: Analyzing the second equality.**

Now we analyze the second part:
\[ \frac{a_2 + a_3}{a_1 + a_4} = 3 \cdot \frac{a_2 - a_3}{a_1 - a_4} \]

Cross-multiplying gives:
\[ (a_2 + a_3)(a_1 - a_4) = 3(a_2 - a_3)(a_1 + a_4) \]

Expanding both sides:
\[ a_2 a_1 - a_2 a_4 + a_3 a_1 - a_3 a_4 = 3(a_2 a_1 + a_2 a_4 - a_3 a_1 - a_3 a_4) \]

4. **Step 4: Rearranging the second equation.**

Rearranging gives:
\[ a_2 a_1 - a_2 a_4 + a_3 a_1 - a_3 a_4 - 3a_2 a_1 - 3a_2 a_4 + 3a_3 a_1 + 3a_3 a_4 = 0 \]

Combining like terms:
\[ -2a_2 a_1 + 4a_3 a_1 + 2a_3 a_4 - 4a_2 a_4 = 0 \]

5. **Step 5: Establishing the relationship.**

From the rearranged equations, we can find that:
\[ \frac{1}{a_4} - \frac{1}{a_3} = \frac{1}{a_2} - \frac{1}{a_1} \]

This indicates that \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \frac{1}{a_4} \) are in Arithmetic Progression (AP).

6. **Step 6: Conclusion.**

Since the reciprocals of \( a_1, a_2, a_3, a_4 \) are in AP, it follows that \( a_1, a_2, a_3, a_4 \) are in Harmonic Progression (HP).

Thus, the final answer is that \( a_1, a_2, a_3, a_4 \) are in Harmonic Progression (HP).

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