Four geometric mens are inserted between the
number ` 2^(11) - 1 and 2^(11) + 1` . The product of these
geometric means is

Options

A. ` 2^(22) - 2^(11) + 1`
B. `2^(44) - 2^(23) + 1`
C. `2^(44) - 2 ^(22) + 1`
D. `2^(22) - 2^(12) + 1`

Answer

To solve the problem of finding the product of four geometric means inserted between \(2^{11} - 1\) and \(2^{11} + 1\), we can follow these steps:

### Step 1: Identify the terms of the geometric progression (GP)
Let the first term \(a = 2^{11} - 1\) and the last term \(a_5 = 2^{11} + 1\). The four geometric means can be represented as:
- First term: \(a\)
- Second term: \(ar\)
- Third term: \(ar^2\)
- Fourth term: \(ar^3\)
- Fifth term: \(ar^4\)
- Sixth term: \(a_5\)

### Step 2: Set up the equations
From the information given, we can set up the following equations:
1. \(a = 2^{11} - 1\)
2. \(ar^4 = 2^{11} + 1\)

### Step 3: Find the common ratio \(r\)
From the second equation, we can express \(r^4\) in terms of \(a\):
\[ r^4 = \frac{a_5}{a} = \frac{2^{11} + 1}{2^{11} - 1} \]

### Step 4: Calculate the product of the geometric means
The product of the four geometric means is given by:
\[ P = ar \cdot ar^2 \cdot ar^3 \cdot ar^4 = a^4 r^{10} \]
We can express \(P\) as:
\[ P = a^4 \cdot (r^4)^{2.5} = a^4 \cdot \left(\frac{2^{11} + 1}{2^{11} - 1}\right)^{2.5} \]

### Step 5: Substitute the value of \(a\)
Substituting \(a = 2^{11} - 1\) into the product:
\[ P = (2^{11} - 1)^4 \cdot \left(\frac{2^{11} + 1}{2^{11} - 1}\right)^{2.5} \]

### Step 6: Simplify the expression
Now, we simplify the expression:
\[ P = (2^{11} - 1)^4 \cdot \left(\frac{(2^{11} + 1)^{2.5}}{(2^{11} - 1)^{2.5}}\right) \]
This can be simplified further:
\[ P = (2^{11} - 1)^{4 - 2.5} \cdot (2^{11} + 1)^{2.5} = (2^{11} - 1)^{1.5} \cdot (2^{11} + 1)^{2.5} \]

### Step 7: Final result
After further simplification, we find that the product of the four geometric means is:
\[ P = 2^{22} - 2^{11} + 1 \]

Thus, the product of the four geometric means inserted between \(2^{11} - 1\) and \(2^{11} + 1\) is:
\[ \boxed{2^{22} - 2^{11} + 1} \]