It is infinity. I think we can do it like this:

Total distance travelled by the particle = time for mirrors to collide * velocity of particle

                                                                     =d/2v * 3v = 3d/2

Consider first trip of particle,

It will hit the 2nd mirror at a distance = 3/4 *d     (bcause particle vel is 3v and mirror vel is v)

By this time both the mirrors have travelled a distance of d/2 together. So remaining distance is d/2 only.

Insecond trip it will hit  1st mirror at a distance = 3/4*d/2

In third trip distance travelled by particle is = 3/4*d/4

In  fourth  trip......................................................= 3/4*d/8  and so on....

So the total distance travelled by particle =3/4*d +3/4 *d/2 +3/4 * d/4 + 3/4 * d/8 +.....+....+  ..

This is a G.P , a=3/4*d & r= 1/2

So the sum =  a(1-rⁿ)/(1-r)

                      3/4*d (1-(1/2)ⁿ)/(1-1/2) =3d/2 (taken from the above)

                                         1-(1/2)ⁿ = 1

                                             (1/2)ⁿ = 0

                                             1/2ⁿ =0 This is possible only when n= (infinity)