The fourth term of the sequence `1/(1+sqrt(x)),1/(1-x),1/(1-sqrt(x)), .....` is:

Options

A. `1/(1-2sqrt(x))`
B. `1/(2-sqrt(x))`
C. `(1+2sqrt(x))/(1-x)`
D. none of these

Answer

To find the fourth term of the sequence given by \( \frac{1}{1+\sqrt{x}}, \frac{1}{1-x}, \frac{1}{1-\sqrt{x}} \), we will first check if these terms are in Arithmetic Progression (AP).

### Step 1: Identify the terms
Let:
- \( A = \frac{1}{1+\sqrt{x}} \) (first term)
- \( B = \frac{1}{1-x} \) (second term)
- \( C = \frac{1}{1-\sqrt{x}} \) (third term)

### Step 2: Check if the terms are in AP
For the terms to be in AP, the following condition must hold:
\[ 2B = A + C \]
Substituting the values of \( A \), \( B \), and \( C \):
\[ 2 \cdot \frac{1}{1-x} = \frac{1}{1+\sqrt{x}} + \frac{1}{1-\sqrt{x}} \]

### Step 3: Simplify the right-hand side
To simplify \( \frac{1}{1+\sqrt{x}} + \frac{1}{1-\sqrt{x}} \), we need a common denominator:
\[ \frac{1}{1+\sqrt{x}} + \frac{1}{1-\sqrt{x}} = \frac{(1-\sqrt{x}) + (1+\sqrt{x})}{(1+\sqrt{x})(1-\sqrt{x})} \]
This simplifies to:
\[ \frac{2}{1 - (\sqrt{x})^2} = \frac{2}{1 - x} \]

### Step 4: Set the left-hand side equal to the right-hand side
Now we have:
\[ 2 \cdot \frac{1}{1-x} = \frac{2}{1-x} \]
This confirms that the terms are indeed in AP.

### Step 5: Find the common difference \( D \)
The common difference \( D \) can be calculated as:
\[ D = B - A = \frac{1}{1-x} - \frac{1}{1+\sqrt{x}} \]
Finding a common denominator:
\[ D = \frac{(1+\sqrt{x}) - (1-x)}{(1-x)(1+\sqrt{x})} = \frac{\sqrt{x} + x}{(1-x)(1+\sqrt{x})} \]

### Step 6: Calculate the fourth term
The fourth term \( T_4 \) can be calculated as:
\[ T_4 = C + D = \frac{1}{1-\sqrt{x}} + D \]
Substituting \( D \):
\[ T_4 = \frac{1}{1-\sqrt{x}} + \frac{\sqrt{x} + x}{(1-x)(1+\sqrt{x})} \]

### Step 7: Combine the terms
To combine these fractions, we need a common denominator:
\[ T_4 = \frac{(1-x)(1+\sqrt{x}) + (\sqrt{x} + x)(1-\sqrt{x})}{(1-\sqrt{x})(1-x)(1+\sqrt{x})} \]
After simplifying the numerator, we will arrive at:
\[ T_4 = \frac{1 + 2\sqrt{x}}{1-x} \]

### Final Result
Thus, the fourth term of the sequence is:
\[ \frac{1 + 2\sqrt{x}}{1-x} \]

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