In a resonance tube experiment, three successive resonances are...

In a resonance tube experiment, the difference between successive resonant lengths of the air column is constant and equal to $\dfrac{\lambda}{2}$, where $\lambda$ is the wavelength of the sound wave.

Let the three successive resonant lengths be $L_1$, $L_2$, and $L_3$.

We have:
$L_2 - L_1 = \dfrac{\lambda}{2}$
$L_3 - L_2 = \dfrac{\lambda}{2}$

This implies that the resonant lengths $L_1$, $L_2$, and $L_3$ are in an arithmetic progression (A.P.).

Given that the lengths are in the ratio $1 : 5 : x$, we can assume:
$L_1 = k$
$L_2 = 5k$
$L_3 = xk$

Since they are in A.P., the common difference must be the same:
$L_2 - L_1 = L_3 - L_2$
$5k - k = xk - 5k$
$4k = (x - 5)k$

Dividing by $k$ (since $k \neq 0$):
$4 = x - 5$
$x = 9$

The given values of the tuning fork's frequency and the velocity of sound are extraneous information and are not required to find the value of $x$.

Answer: $9$