设Sn=-1+3-5+7-…+(-1)n(2n-1),则Sn=_.

问题描述:

设Sn=-1+3-5+7-…+(-1)n(2n-1),则Sn=______.

当n是偶数时,Sn=(-1+3)+(-5+7)+…+[-(2n-3)+(2n-1)]=2+2+…+2(共

n
2
项)=
n
2
=n

当n是奇数时,Sn=(-1+3)+(-5+7)+…+[-(2n-5)+(2n-3)]+(-1)×(2n-1)
=2+2+…+2(共有
n−1
2
项)-(2n-1)
=
2×(n−1)
2
−2n+1

=n-1-2n+1
=-n.
∴Sn=(-1)n•n.
故答案为:(-1)n•n.