Answer
$2f_3$
Work Step by Step
A string under tension $\tau_i$ oscillates in the third harmonic at frequency $f_3$, and the waves on the string have wavelength $\lambda_3$
Therefore,
$f_3=\frac{v}{\lambda_3}$
or, $f_3=\frac{1}{\lambda_3}\sqrt {\frac{\tau_i}{\mu}}$
or, $\lambda_3=\frac{1}{f_3}\sqrt {\frac{\tau_i}{\mu}}$
If the tension is increased to $\tau_f=4\tau_i$ and the string is again made to oscillate in the third harmonic, then the frequency of oscillation is given by
$f_3^{'}=\frac{1}{\lambda_3}\sqrt {\frac{\tau_f}{\mu}}$
$f_3^{'}=\frac{1}{\frac{1}{f_3}\sqrt {\frac{\tau_i}{\mu}}}\sqrt {\frac{4\tau_i}{\mu}}$
or, $f_3^{'}=2f_3$
$\therefore$ The frequency of oscillation is $2f_3$
