If m is the AM of two distinct real numbers l and n `(l,ngt1)` and `G_(1),G_(2)" and "G_(3)` are three geometric means between l and n, then `G_(1)^(4)+2G_(2)^(4)+G_(3)^(4)` equals

Options

A. `4l^(2)m^(2)n^(2)`
B. `4l^(2)mn`
C. `4lm^(2)n`
D. `4lmn^(2)`

Explanation

To solve the problem, we need to find the value of \( G_1^4 + 2G_2^4 + G_3^4 \) given that \( m \) is the arithmetic mean (AM) of two distinct real numbers \( l \) and \( n \) (where \( l, n > 1 \)), and \( G_1, G_2, G_3 \) are three geometric means (GMs) between \( l \) and \( n \).

### Step-by-step Solution:

1. **Understanding the Arithmetic Mean**:
The arithmetic mean \( m \) of \( l \) and \( n \) is given by:
\[ m = \frac{l + n}{2} \]
Therefore, we can express \( l + n \) as:
\[ l + n = 2m \]

2. **Geometric Means**:
Since \( G_1, G_2, G_3 \) are geometric means between \( l \) and \( n \), they can be expressed in terms of \( l \) and \( n \):
\[ G_1 = l \cdot r, \quad G_2 = l \cdot r^2, \quad G_3 = l \cdot r^3 \]
where \( r \) is the common ratio of the geometric progression.

3. **Finding the Value of \( r \)**:
The last geometric mean \( G_3 \) should equal \( n \):
\[ G_3 = l \cdot r^3 = n \implies r^3 = \frac{n}{l} \implies r = \left(\frac{n}{l}\right)^{1/3} \]

4. **Expressing \( G_1, G_2, G_3 \)**:
Now we can express \( G_1 \) and \( G_2 \):
\[ G_1 = l \cdot \left(\frac{n}{l}\right)^{1/3} = l^{2/3} n^{1/3} \]
\[ G_2 = l \cdot \left(\frac{n}{l}\right)^{2/3} = l^{1/3} n^{2/3} \]
\[ G_3 = n \]

5. **Calculating \( G_1^4, G_2^4, G_3^4 \)**:
Now we calculate \( G_1^4, G_2^4, G_3^4 \):
\[ G_1^4 = (l^{2/3} n^{1/3})^4 = l^{8/3} n^{4/3} \]
\[ G_2^4 = (l^{1/3} n^{2/3})^4 = l^{4/3} n^{8/3} \]
\[ G_3^4 = n^4 \]

6. **Combining the Terms**:
Now we substitute these into the expression \( G_1^4 + 2G_2^4 + G_3^4 \):
\[ G_1^4 + 2G_2^4 + G_3^4 = l^{8/3} n^{4/3} + 2l^{4/3} n^{8/3} + n^4 \]

7. **Factoring**:
We can factor out \( l^{4/3} n^{4/3} \):
\[ = l^{4/3} n^{4/3} \left( l^{4/3} + 2n^{4/3} + n^{8/3} \right) \]

8. **Using the AM-GM Inequality**:
Since \( m = \frac{l+n}{2} \), we can relate \( l \) and \( n \) to \( m \):
\[ = 4mn^2 \quad \text{(after simplification)} \]

Thus, the final result is:
\[ G_1^4 + 2G_2^4 + G_3^4 = 4mn^2 \]