The force on the particle is given by F=qvB when the motion is perpendicular to the field. Here q is the electric charge and B is the magnetic field in tesla. The resulting motion is circular. According to Newton's laws of motion an object in circular motion is accelerating radially according to a = v²/r where r is the radius of the circular path. Since F=ma, we can use this acceleration to get the radial force, which is F=mv²/r. This force and the force from the magnetic field are equivalent so
mv²/r=qvB. The particle's velocity is then
v=qBr/m
Note the distance around the circle is 2πr. This is a linear function in r, so if you double r, you double the length around the circle. Since the velocity equation is directly proportional to r, if you triple v, you must also triple the right hand side of the equation as well. But on the right side of the equation, q, B, and m are constant. The only thing you can triple is r, meaning that while the particle is moving 3 times as fast as before, it now has 3 times further to travel, meaning
the time to go around is the same. As the particle speeds up, the circle it is moving in gets bigger. To change the time to go around, you'd need to change the magnetic field.
