设数列{an}满足当n=2k-1(k∈N﹢)时 ,an=n,当;当n=2k(k∈N*)时,an=ak.,记Sn=a1+a2+a3……+a2n-1+a2n(1)求S3 (2)证明Sn=4^n-1+Sn-1(n>=2)

问题描述:

设数列{an}满足当n=2k-1(k∈N﹢)时 ,an=n,当;当n=2k(k∈N*)时,an=ak.,记Sn=a1+a2+a3……+a2n-1+a2n(1)求S3 (2)证明Sn=4^n-1+Sn-1(n>=2)
证明1/S1+1/S2+……+Sn-1

证明如下:(1)S3=a1+a2+a3+a4+a5+a6+a7+a8=a1+a1+a3+a1+a5+a3+a7+a1=4a1+2a3+a5+a7=4×1+2×3+5+7=22(2)Sn=a1+a2+…+a2n-1+a2n=(a1+a3+…+a2n-1)+(a2+a4+…+a2n)=[1+3+…+(2n-1)]+(a2+a4+a6+…+a2n)=4n-1+(a1+...