If `x_1,x_2 & x_3` are the three real solutions of the equation `x^(log_10^2 x+log_10 x^3+3)=2/((1/(sqrt(x+1)-1))-(1/(sqrt(x+1)+1))),` where `x_1 > x_2 > x_3,`
then these are in

Options

A. AP
B. GP
C. HP
D. `a^-1+b^-1=c^-1`

Answer

To solve the equation given in the problem, we will follow these steps systematically:

### Step 1: Write down the given equation
The equation is:
\[ x^{(\log_{10}^2 x + \log_{10} x^3 + 3)} = \frac{2}{\left(\frac{1}{\sqrt{x+1}-1}\right) - \left(\frac{1}{\sqrt{x+1}+1}\right)} \]

### Step 2: Simplify the right-hand side
To simplify the right-hand side, we first find a common denominator for the fractions:
\[ \frac{1}{\sqrt{x+1}-1} - \frac{1}{\sqrt{x+1}+1} \]
The common denominator is \((\sqrt{x+1}-1)(\sqrt{x+1}+1)\), which simplifies to:
\[ (\sqrt{x+1})^2 - 1^2 = x + 1 - 1 = x \]
Thus, the right-hand side becomes:
\[ \frac{2}{\frac{2}{x}} = x \]

### Step 3: Set the left-hand side equal to the simplified right-hand side
Now we have:
\[ x^{(\log_{10}^2 x + \log_{10} x^3 + 3)} = x \]

### Step 4: Take logarithm on both sides
Taking logarithm base 10 on both sides gives us:
\[ \log_{10}\left(x^{(\log_{10}^2 x + \log_{10} x^3 + 3)}\right) = \log_{10} x \]
Using the property \(\log(a^b) = b \cdot \log a\), we can rewrite the left side:
\[ (\log_{10}^2 x + \log_{10} x^3 + 3) \cdot \log_{10} x = \log_{10} x \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ \log_{10}^2 x + 3\log_{10} x + 3 - 1 = 0 \]
This simplifies to:
\[ \log_{10}^2 x + 3\log_{10} x + 2 = 0 \]

### Step 6: Let \(y = \log_{10} x\)
Substituting \(y = \log_{10} x\), we have:
\[ y^2 + 3y + 2 = 0 \]

### Step 7: Factor the quadratic equation
Factoring gives:
\[ (y + 1)(y + 2) = 0 \]
Thus, the solutions for \(y\) are:
\[ y = -1 \quad \text{or} \quad y = -2 \]

### Step 8: Convert back to \(x\)
Now converting back to \(x\):
1. For \(y = -1\):
\[ \log_{10} x = -1 \implies x = 10^{-1} = \frac{1}{10} \]
2. For \(y = -2\):
\[ \log_{10} x = -2 \implies x = 10^{-2} = \frac{1}{100} \]

### Step 9: Include the solution \(y = 0\)
Additionally, \(y = 0\) gives:
\[ \log_{10} x = 0 \implies x = 10^0 = 1 \]

### Step 10: List the solutions
The three solutions are:
\[ x_1 = 1, \quad x_2 = \frac{1}{10}, \quad x_3 = \frac{1}{100} \]
Ordering them gives:
\[ x_1 > x_2 > x_3 \]

### Step 11: Check for Arithmetic Progression, Geometric Progression, and Harmonic Progression
To check if these values are in AP, GP, or HP:
- **AP**: The differences are not constant.
- **GP**: The ratio is constant:
\[ \frac{x_2}{x_1} = \frac{1/10}{1} = \frac{1}{10}, \quad \frac{x_3}{x_2} = \frac{1/100}{1/10} = \frac{1}{10} \]
- **HP**: The reciprocals are in AP:
\[ \frac{1}{x_1} = 1, \quad \frac{1}{x_2} = 10, \quad \frac{1}{x_3} = 100 \]
The differences \(10 - 1 = 9\) and \(100 - 10 = 90\) are not equal.

Thus, the solutions \(x_1, x_2, x_3\) are in **Geometric Progression (GP)**.