Finitely generated not implies residually finite

Statement
A finitely generated group need not be a residually finite group. In particular, the notion of finitely generated residually finite group is strictly stronger than the notion of finitely generated group.

Related facts

 * Finitely generated abelian implies residually finite
 * Free implies residually finite
 * Finitely generated and residually finite implies Hopfian

Any finitely generated simple group
Consider any infinite finitely generated simple group. Since this has no proper nontrivial normal subgroup, it clearly does not have any normal subgroup of finite index, hence it cannot be residually finite. On the other hand, by construction, it is finitely generated.