数列求和已知数列『An]为等比数列,其首项为0.5,公比为0.5.前n项和为Sn,则数列nSn]的前n项和Tn.

Question

数列求和
已知数列『An]为等比数列,其首项为0.5,公比为0.5.前n项和为Sn,则数列【nSn]的前n项和Tn.

Answer 1

Sn=0.5×(1-0.5^n)/(1-0.5)=1-0.5^n
nSn=n-n×0.5^n
数列分为两部分,前半部分{n}是等差数列,后半部分{-n×0.5^n}是等差数列和等比数列相乘.
数列{n}前n项和
Tn1=(1+n)n/2
数列{n×0.5^n}前n项和
Tn2=1×0.5^1+2×0.5^2+3×0.5^3+……+n×0.5^n ①
两边同乘公比倒数2
2Tn2=1×0.5^0+2×0.5^1+3×0.5^2+……+n×0.5^(n-1) ②
②-①得
Tn2=1×0.5^0+[0.5^1+0.5^2+0.5^3+……+0.5^(n-1)]-n×0.5^n
中括号内是等比数列,共n-1项
Tn2=1+0.5×(1-0.5^(n-1))/(1-0.5)-n×0.5^n
=2-(2+n)×0.5^n
Tn=Tn1-Tn2
=(1+n)n/2-2+(2+n)×0.5^n