解：?$(1)①S_2-S_1=(1+2\sqrt {2})2-(1+\sqrt {2})2=6+2\sqrt {2}$?
?$②S_3-S_2=(1+3\sqrt {2})2-(1+2\sqrt {2})2=10+2\sqrt {2}$?
?$③S_4-S_3=(1+4\sqrt {2})2-(1+3\sqrt {2})2=14+2\sqrt {2}$?
?$(2)S_{n+1}-S_{n}=2(2n+1)+2\sqrt {2}$?
?$S_{n+1}-S_{n}=[1+(n+1)\sqrt {2}]2-(1+\sqrt {2}n)2$?
?$=1+2\sqrt {2}(n+1)+2(n+1)-1-2\sqrt {2}n-2n2$?
?$=2(2n+1)+2\sqrt {2}$?
