The sequence `[x_n]` is a GP with `x_2/x_4 `= 1/4 and `x_1 + x_4 = 108`. What will be the value of `x_3` ?

Options

A. (A) 42
B. (B) 48
C. (C) 44
D. (D) 56

Answer

To solve the problem, we need to find the value of \( x_3 \) in the geometric progression (GP) defined by the conditions given in the question. Let's break it down step by step.

### Step 1: Understand the terms of the GP
In a geometric progression, the \( n \)-th term can be expressed as:
\[ x_n = x_1 \cdot r^{n-1} \]
where \( x_1 \) is the first term and \( r \) is the common ratio.

### Step 2: Set up the equations based on the problem
We are given two conditions:
1. \( \frac{x_2}{x_4} = \frac{1}{4} \)
2. \( x_1 + x_4 = 108 \)

Using the formula for the terms of the GP:
- The second term \( x_2 = x_1 \cdot r^{2-1} = x_1 \cdot r \)
- The fourth term \( x_4 = x_1 \cdot r^{4-1} = x_1 \cdot r^3 \)

### Step 3: Substitute into the first equation
Substituting \( x_2 \) and \( x_4 \) into the first condition:
\[ \frac{x_1 \cdot r}{x_1 \cdot r^3} = \frac{1}{4} \]
This simplifies to:
\[ \frac{1}{r^2} = \frac{1}{4} \]
From this, we can find \( r^2 \):
\[ r^2 = 4 \implies r = 2 \quad (\text{since } r \text{ is positive in a GP}) \]

### Step 4: Substitute into the second equation
Now we substitute \( r = 2 \) into the second condition:
\[ x_1 + x_4 = 108 \]
Substituting \( x_4 \):
\[ x_1 + x_1 \cdot r^3 = 108 \]
This becomes:
\[ x_1 + x_1 \cdot 2^3 = 108 \]
\[ x_1 + 8x_1 = 108 \]
\[ 9x_1 = 108 \]
Dividing both sides by 9:
\[ x_1 = 12 \]

### Step 5: Find \( x_3 \)
Now that we have \( x_1 \) and \( r \), we can find \( x_3 \):
\[ x_3 = x_1 \cdot r^{3-1} = x_1 \cdot r^2 \]
Substituting the values:
\[ x_3 = 12 \cdot 2^2 = 12 \cdot 4 = 48 \]

Thus, the value of \( x_3 \) is \( \boxed{48} \).

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