Three resonant frequencies of a string are measured as 90 Hz , 150...

The given resonant frequencies of the string are $90 \text{ Hz}$, $150 \text{ Hz}$, and $210 \text{ Hz}$.

The fundamental frequency $f_0$ is the highest common factor (HCF) of these frequencies.

$f_0 = \text{HCF}(90, 150, 210) = 30 \text{ Hz}$

The harmonic numbers corresponding to the given frequencies are:

$n_1 = \dfrac{90}{30} = 3$ (3rd harmonic)

$n_2 = \dfrac{150}{30} = 5$ (5th harmonic)

$n_3 = \dfrac{210}{30} = 7$ (7th harmonic)

For a string fixed at both ends, the $n^{\text{th}}$ harmonic corresponds to the $(n-1)^{\text{th}}$ overtone.

Therefore, the 3rd, 5th, and 7th harmonics correspond to the 2nd, 4th, and 6th overtones respectively.