Cross sections and analyzing powers for proton elastic scattering from Sn-116,Sn-118,Sn-120,Sn-122,Sn-124 at 295 MeV have been measured for a momentum transfer of up to about 3.5 fm(-1) to deduce systematic changes of the neutron density distribution. We tuned the relativistic Love-Franey interaction to explain the proton elastic scattering of a nucleus whose density distribution is well known. Then, we applied this interaction to deduce the neutron density distributions of tin isotopes. The result of our analysis shows the clear systematic behavior of a gradual increase in the neutron skin thickness of tin isotopes with mass number.