In any progression, if `(t_(2)t_(3))/(t_(1)t_(4)) = (t_(2) + t_(3))/(t_(1) + t_(4)) = 3 ((t_(2) - t_(3))/(t_(1) - t_(4)))`, then `t_(1), t_(2), t_(3), t_(4)` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. none

Answer

To solve the problem, we need to analyze the given equations step by step. We are given that:

\[ \frac{t_2 t_3}{t_1 t_4} = \frac{t_2 + t_3}{t_1 + t_4} = 3 \cdot \frac{t_2 - t_3}{t_1 - t_4} \]

Let's denote the three expressions as follows:

1. \( A = \frac{t_2 t_3}{t_1 t_4} \)
2. \( B = \frac{t_2 + t_3}{t_1 + t_4} \)
3. \( C = 3 \cdot \frac{t_2 - t_3}{t_1 - t_4} \)

### Step 1: Set \( A = B \)

We start by equating \( A \) and \( B \):

\[ \frac{t_2 t_3}{t_1 t_4} = \frac{t_2 + t_3}{t_1 + t_4} \]

Cross-multiplying gives:

\[ t_2 t_3 (t_1 + t_4) = (t_2 + t_3) t_1 t_4 \]

Expanding both sides:

\[ t_2 t_3 t_1 + t_2 t_3 t_4 = t_1 t_4 t_2 + t_1 t_4 t_3 \]

Rearranging terms:

\[ t_2 t_3 t_1 + t_2 t_3 t_4 - t_1 t_4 t_2 - t_1 t_4 t_3 = 0 \]

### Step 2: Factor the equation

We can factor out common terms:

\[ t_2 t_3 t_1 + t_2 t_3 t_4 - t_1 t_4 t_2 - t_1 t_4 t_3 = 0 \]

This simplifies to:

\[ t_2 t_3 (t_1 + t_4) = t_1 t_4 (t_2 + t_3) \]

### Step 3: Set \( A = C \)

Now we set \( A \) equal to \( C \):

\[ \frac{t_2 t_3}{t_1 t_4} = 3 \cdot \frac{t_2 - t_3}{t_1 - t_4} \]

Cross-multiplying gives:

\[ t_2 t_3 (t_1 - t_4) = 3 t_1 t_4 (t_2 - t_3) \]

Expanding both sides:

\[ t_2 t_3 t_1 - t_2 t_3 t_4 = 3 t_1 t_4 t_2 - 3 t_1 t_4 t_3 \]

Rearranging terms:

\[ t_2 t_3 t_1 + 3 t_1 t_4 t_3 = 3 t_1 t_4 t_2 + t_2 t_3 t_4 \]

### Step 4: Combine and simplify

Now we have two equations from steps 1 and 3. We can combine them to analyze the relationships between \( t_1, t_2, t_3, \) and \( t_4 \).

### Step 5: Conclude the relationship

From the equations derived, we can conclude that \( t_1, t_2, t_3, t_4 \) must satisfy the condition of being in Harmonic Progression (HP).

Thus, the final conclusion is:

\[ t_1, t_2, t_3, t_4 \text{ are in Harmonic Progression (HP)} \]