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=______.
答
当n是偶数时,Sn=(-1+3)+(-5+7)+…+[-(2n-3)+(2n-1)]=2+2+…+2(共
项)=2×n 2
=n.n 2
当n是奇数时,Sn=(-1+3)+(-5+7)+…+[-(2n-5)+(2n-3)]+(-1)×(2n-1)
=2+2+…+2(共有
项)-(2n-1)n−1 2
=
−2n+12×(n−1) 2
=n-1-2n+1
=-n.
∴Sn=(-1)n•n.
故答案为:(-1)n•n.