设数列〔an〕满足a1=1，a2=5/3(3分之5），an+2=5/3an+1－2/3an,(n属于N※）. 求数列{nan}的前n项和Sn
(1)
a(n+2)=(5/3)a(n+1)-(2/3)an
a(n+2)-a(n+1) = (2/3)(a(n+1) -an)
{a(n+1) -an} 是等比数列, q=2/3
a(n+1) -an = (2/3)^(n-1).(a2-a1)
= (2/3)^n
bn = a(n+1) -an = (2/3)^n
a(n+1) - an = (2/3)^n
an -a(n-1) = (2/3)^(n-1)
an - a1 = (2/3)+(2/3)^2+....+(2/3)^(n-1)
= (2/3)(1- (2/3)^(n-1) )/(1-2/3)
= 2(1- (2/3)^(n-1) )
an = 3 - 2(2/3)^(n-1)
(2)
nan = 3n - 2[n(2/3)^(n-1)]
let
S = 1.(2/3)^0+2.(2/3)^1+...+n(2/3)^(n-1) (1)
(2/3)S = 1.(2/3)^1+2.(2/3)^2+...+n(2/3)^n (2)
(1)-(2)
(1/3)S = [1+2/3+...+(2/3)^(n-1)]- n(2/3)^n
= 3[1- (2/3)^n] - n(2/3)^n
S =9[1- (2/3)^n] - 3n(2/3)^n
= 9 - (3n-9)(2/3)^n
Sn = 3n(n+1)/2 - 2S
=n(n+1) - 18 + 2(3n-9)(2/3)^n
(2)
an-a(n-1)=(2/3)^(n-1)
a(n-1)-a(n-2)=(2/3)^(n-2)
.....
a3-a2=(2/3)^2
a2-a1=(2/3)^1
各等式相加 所以
an-a1=(2/3)^1+(2/3)^2+…+(2/3)^(n-1)
an=a1+(2/3)*(1-(2/3)^(n-1))/(1-2/3)
=3-2(2/3)^(n-1)
所以 Sn=6(2/3)^n+3n-6
a(n+2)=(5/3)a(n+1)-(2/3)an
a(n+2)-a(n+1)=(2/3)a(n+1)-(2/3)an=(2/3)[a(n+1)-an]
[a(n+2)-a(n+1)]/[a(n+1)-an]=2/3，为定值。
a2-a1=5/3 -1=2/3
数列{a(n+1)-an}是以2/3为首项，2/3为公比的等比数列。
a(n+1)-an=(2/3)×(2/3)^(n-1)=(2/3)ⁿ
an-a(n-1)=(2/3)^(n-1)
a(n-1)-a(n-2)=(2/3)^(n-2)
…………
a2-a1=2/3
累加
an -a1=2/3 +(2/3)²+...+(2/3)^(n-1)
an=a1+2/3 +(2/3)²+...+(2/3)^(n-1)
=1+2/3 +(2/3)²+...+(2/3)^(n-1)
=1×[1-(2/3)ⁿ]/(1-2/3)
=3 -3×(2/3)ⁿ
数列{an}的通项公式为an=3-3×(2/3)ⁿ
nan=n×[3-3×(2/3)ⁿ]=3n -3×[n×(2/3)ⁿ]
Sn=a1+2a2+3a3+...+nan
=3(1+2+...+n) -3×[1×(2/3)+2×(2/3)²+3×(2/3)³+...+n×(2/3)ⁿ]
令Cn=1×(2/3)+2×(2/3)²+3×(2/3)³+...+n×(2/3)ⁿ
则(2/3)Cn=1×(2/3)²+2×(2/3)³+...+(n-1)×(2/3)ⁿ+n×(2/3)^(n+1)
Cn-(2/3)Cn=(1/3)Cn
=(2/3)+(2/3)²+...+(2/3)ⁿ -n×(2/3)^(n+1)
=(2/3)[1-(2/3)ⁿ]/(1-2/3) -n×(2/3)^(n+1)
=2 -(n+2)×(2/3)^(n+1)
Cn=6-(2n+4)×(2/3)ⁿ
Sn=3(1+2+...+n)-3Cn
=3n(n+1)/2 -18 -(n+2)×2^(n+1) /3^(n-1)
设bn=an+1-an
bn+1=2/3bn
bn=(2/3)^n=an+1-an
...................
3a(n+2)=5a(n+1)-2an3a(n+2)-3a(n+1)=2a(n+1)-2an[a(n+2)-a(n+1)]/[a(n+1)-an]=2/3a(n+1)-an=(a2-a1)(2/3)^(n-1)=(2/3)^na2-a1=2/3a3-a2=(2/3)^2a4-a3=(2/3)^3……a(n-1)-a(n-2)=(2/3)^(n-2)an-a(n-1)=(2/3)^(n-1)将上列式子相加得an-a1=2/3+(2/3)^2+(2/3)^3+……(2/3)^(n-1)=(2/3)[1-(2/3)^(n-1)]/(1-2/3)an=3-2(2/3)^(n-1)nan=3n-2n(2/3)^(n-1)Sn=3(1+2+3+……+n)-2[1+2*(2/3)+3*(2/3)^2+……+(n-1)(2/3)^(n-2)+n(2/3)^(n-1)]=3n(n+1)/2-2[1+2*(2/3)+3*(2/3)^2+……+(n-1)(2/3)^(n-2)+n(2/3)^(n-1)](2/3)Sn=n(n+1)-2[(2/3)+2*(2/3)^2+……+(n-2)(2/3)^(n-2)+(n-1)(2/3)^(n-1)+n(2/3)^n]Sn-(2/3)Sn=Sn/3=n(n+1)/2-2[1-n(2/3)^n+(2/3)+(2/3)^2+……+(2/3)^(n-2)+(2/3)^(n-1)]=n(n+1)/2-2[1-n(2/3)^n+(2/3)(1-(2/3)^(n-1))/(1-2/3)]=n(n+1)/2-2[1-n(2/3)^n+2-(2/3)^n]=n(n+1)/2+2(n+1)(2/3)^n-6Sn=3n(n+1)/2+6(n+1)(2/3)^n-18
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