已知an,bn满足a1=2,b1=1且an=3/4a(n-1)+1/4b(n-1)+1,bn=1/4a(n-1)+3/4b(n-1)+1(a>=2) 令cn=an+bn 设数列Cn的前n和为sn,求证1/S1+1/S2+1/S3+.+1/S
问题描述:
已知an,bn满足a1=2,b1=1且an=3/4a(n-1)+1/4b(n-1)+1,bn=1/4a(n-1)+3/4b(n-1)+1(a>=2) 令cn=an+bn 设数列Cn的前n和为sn,求证1/S1+1/S2+1/S3+.+1/Sn
n要大于等于2
答
Cn=an+bn=[3/4a(n-1)+1/4b(n-1)+1] + [1/4a(n-1)+3/4b(n-1)+1]= a(n-1)+b(n-1) +2=C(n-1) +2C1=a1+b1=3所以Cn是等差数列,Cn=2n+1Sn= 3+5+7+……+(2n+1)= n+n(n+1) = n(n+2)1/Sn= 1/[n(n+2)]= [1/n - 1/(n+...