If `n_1,n_2` and `n_3` are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency `n` of the string is given by

Options

A. `(1)/(n)=(1)/(n_1)+(1)/(n_2)+(1)/(n_3)`
B. `(1)/(sqrtn)=(1)/(sqrtn_1)+(1)/(sqrtn_2)+(1)/(sqrtn_3)`
C. `sqrtn=sqrtn_1+sqrtn_2+sqrtn_3`
D. `n=n_1+n_2+n_3``

Explanation

To determine the original fundamental frequency \( n \) of a string that has been divided into three segments with fundamental frequencies \( n_1, n_2, \) and \( n_3 \), we can use the relationship between frequency, tension, linear mass density, and length of the string segments.

### Step-by-Step Solution:

1. **Understand the relationship of frequency with tension and mass density**:
The fundamental frequency \( n \) of a vibrating string is given by the formula:
\[ n = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where \( L \) is the length of the string, \( T \) is the tension, and \( \mu \) is the linear mass density of the string.

2. **Express the frequencies of the segments**:
For each segment of the string, we can express the fundamental frequencies \( n_1, n_2, \) and \( n_3 \) as:
\[ n_1 = \frac{1}{2L_1} \sqrt{\frac{T}{\mu}}, \quad n_2 = \frac{1}{2L_2} \sqrt{\frac{T}{\mu}}, \quad n_3 = \frac{1}{2L_3} \sqrt{\frac{T}{\mu}} \]

3. **Relate the lengths of the segments**:
The total length of the original string \( L \) can be expressed as:
\[ L = L_1 + L_2 + L_3 \]

4. **Find the original frequency \( n \)**:
The original frequency \( n \) of the entire string can be expressed as:
\[ n = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
Substituting \( L \) from the previous step, we get:
\[ n = \frac{1}{2(L_1 + L_2 + L_3)} \sqrt{\frac{T}{\mu}} \]

5. **Combine the expressions for \( n_1, n_2, n_3 \)**:
We can express \( \sqrt{\frac{T}{\mu}} \) in terms of \( n_1, n_2, \) and \( n_3 \):
\[ \sqrt{\frac{T}{\mu}} = 2L_1 n_1 = 2L_2 n_2 = 2L_3 n_3 \]
Hence, we can express \( n \) in terms of the segments:
\[ n = \frac{n_1 L_1 + n_2 L_2 + n_3 L_3}{L_1 + L_2 + L_3} \]

6. **Final expression**:
The original fundamental frequency \( n \) can be expressed as:
\[ n = \frac{n_1 L_1 + n_2 L_2 + n_3 L_3}{L_1 + L_2 + L_3} \]