n=1時, (1-x)(1+x)=1-x^2命題成立.
設(shè)n=k時命題成立,
即有:(1-x)(1+x+x^2+……+x^(k-1)）=1-x^k,
則當(dāng)n=k+1時,有:
(1-x)(1+x+x^2+……+x^(k-1)+x^k)=
=(1-x)(1+x+x^2+……+x^(k-1))+(1-x)*x^k
=1-x^k+(1-x)*x^k
=1-x^k+x^k-x^(k+1)
=1-x^(k+1),
知命題仍成立.
由數(shù)學(xué)歸納法知,此命題對任何正整數(shù)成立.