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Step-by-step Solution
Step 1: Interpret the given distances
We are told that the distance between two consecutive crests (peaks) of a transverse wave is 5 m. This implies that if $n_2$ crests lie between, then the spacing between two crests is given by
$$
5 = n_2\,\lambda,
$$
where $\lambda$ is the wavelength and $n_2$ is the number of whole wavelengths between those two crests.
Step 2: Relate the distance between a crest and a trough
The distance between a crest and a trough is 1.5 m. In terms of the wavelength, the distance between a crest and the next trough can be expressed as
$$
1.5 = \left(2 n_1 + 1\right)\frac{\lambda}{2},
$$
where $n_1$ is an integer that counts how many “half-wavelength” segments plus half more (since a crest to a trough is half a wavelength, plus any integer number of full wavelengths).
Step 3: Form the ratio and simplify
Divide the crest-to-trough equation by the crest-to-crest equation:
$$
\frac{1.5}{5}
= \frac{\left(2 n_1 + 1\right)\frac{\lambda}{2}}{n_2\,\lambda}
= \frac{2 n_1 + 1}{2 n_2}.
$$
Simplifying, we get
$$
\frac{1.5}{5}
= 0.3
= \frac{2 n_1 + 1}{2 n_2}.
$$
Step 4: Convert to an integer relation
Multiply both sides by $2 n_2$:
$$
0.3 \times (2 n_2) = 2 n_1 + 1
\quad\Longrightarrow\quad
0.6\,n_2 = 2 n_1 + 1.
$$
Multiply everything by 5 to clear the decimal:
$$
3 n_2 = 10 n_1 + 5.
$$
Step 5: Find integer solutions
We look for integer solutions $(n_1, n_2)$ that satisfy:
$$
3 n_2 = 10 n_1 + 5.
$$
∘ One solution: $n_1 = 1$, $n_2 = 5$. Then,
$$
\lambda = \frac{5}{n_2} = \frac{5}{5} = 1.
$$
∘ Another solution: $n_1 = 4$, $n_2 = 15$. Then,
$$
\lambda = \frac{5}{n_2} = \frac{5}{15} = \frac{1}{3}.
$$
∘ Another solution: $n_1 = 7$, $n_2 = 25$. Then,
$$
\lambda = \frac{5}{25} = \frac{1}{5}.
$$
And so on, following the linear pattern derived from the integer solutions.
Step 6: Conclude the possible wavelengths
From these integer solutions, the possible values for $ \lambda $ are
$$
1, \frac{1}{3}, \frac{1}{5}, \dots
$$
Thus, the correct option is
$$
\frac{1}{1}, \frac{1}{3}, \frac{1}{5}, \dots
$$
