1+a+a(1+a)+a(1+a)^2+a(1+a)^3+……+a(1+a)^2009
=(1+a)+a(1+a)[(1+a)^2009-1]/(1+a-1)
=(1+a)+(1+a)[(1+a)^2009-1]
=(1+a)[1+(1+a)^2009-1]
=(1+a)^2010
S1=x^2 S2=y^2 S3=(x+y)^2
2(S1+S2)S3-(S1-S2)^2
=2(x^2+y^2)(x+y)^2-(x^2-y^2)^2
=2(x^2+y^2)(x+y)^2-(x^2+y^2)^2+4x^2y^2
=(x^2+y^2)[2(x+y)^2-(x^2+y^2)]+4x^2y^2
=(x^2+y^2)(x^2+4xy+y^2)+4x^2y^2
=(x^2+y^2)^2+4xy(x^2+y^2)+4x^2y^2
=(x^2+y^2+2xy)^2
=(x+y)^4