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Step-by-Step Solution
Step 1: Express the Fundamental Frequency in Terms of String Length
The fundamental frequency $v$ of a stretched string of length $l$, under tension $T$, and having linear mass density $\mu$, is given by
$$
v = \frac{1}{2l} \sqrt{\frac{T}{\mu}}.
$$
Here, $T$ and $\mu$ are constants. Hence, the fundamental frequency is inversely proportional to the length of the string:
$$
v \propto \frac{1}{l}.
$$
Step 2: Relate the Segments' Frequencies to Their Lengths
Let the original length of the string be $l$. If the string is divided into three segments of lengths $l_1$, $l_2$, and $l_3$, their fundamental frequencies are $v_1$, $v_2$, and $v_3$ respectively. Since each frequency is inversely proportional to its corresponding segment length, we can write:
$$
l_1 = \frac{k}{v_1}, \quad l_2 = \frac{k}{v_2}, \quad l_3 = \frac{k}{v_3},
$$
where $k$ is a constant of proportionality (related to $2 \sqrt{\frac{T}{\mu}}$, but its exact form is not needed here). Similarly, the total length $l$ can be written as
$$
l = \frac{k}{v}.
$$
Step 3: Use the Total Length Condition
Since the original string is now divided into three segments,
$$
l = l_1 + l_2 + l_3.
$$
Substituting the expressions for $l_1$, $l_2$, $l_3$, and $l$ in terms of $v_1$, $v_2$, $v_3$, and $v$, we get:
$$
\frac{k}{v} = \frac{k}{v_1} + \frac{k}{v_2} + \frac{k}{v_3}.
$$
Step 4: Simplify to Obtain the Required Relation
Dividing through by $k$ (which is a common, nonzero constant), we arrive at the relation:
$$
\frac{1}{v} = \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}.
$$
This matches the correct option given in the problem statement.
