已知数列an满足sn=2n-an

如图

(1)a1=S1=3-1=2n>1时,an=Sn-S(n-1)=3*(3/2)^(n-2)*(3/2-1)=(3/2)^(n-1)n=1不符合此式,故an=2,n=1an=(3/2)^(n-1),n>

Sn=a1(q^n-1)/(q-1)=a1q^n/(q-1)-a1/(q-1)=b*2^n+a根据等式左右两边相等,得q=2,a1/(q-1)=b,-a1/(q-1)=a所以应满足a+b=0

（Ⅰ）证明：由a1+s1=2a1=2得a1=1；由an+Sn=2n得an+1+Sn+1=2（n+1）两式相减得2an+1-an=2，即2an+1-4=an-2，即an+1-2=12（an-2）是首项为

2a[n]-n-1=a[n-1]【1】待定系数：2(a[n]+xn+y)=a[n-1]+x(n-1)+y【2】将【1】式a[n-1]代入上式：（注意：也可变换后用a[n]代入上式,看方便确定）2(a[

Sn=2n-an,(1)S(n+1)=2*(n+1)-a(n+1)(2)(2)-(1)得:a(n+1)=2-a(n+1)+an.即:2*a(n+1)=2+an.变形:2*[a(n+1)-2]=an-2

n=1时,2a1=2S1=a1^2+1-4a1^2-2a1-3=0(a1+1)(a1-3)=0a1=-1(数列各项均为正,舍去)或a1=3n≥2时,2an=2Sn-2S(n-1)=an^2+n-4-a

因为Sn+Sn-1=3an所以Sn-1+Sn-1+an=3an2Sn-1=2anSn-1=an因为Sn=an+1所以Sn-Sn-1=an+1-anan=an+1-an2an=an+1an+1/an=2

n=1时,a1=S1=a+bn≥2时,Sn=a×n²+bnS(n-1)=a×(n-1)²+b两式相减得：an=Sn-S(n-1)=2a×n-a∴a(n-1)=2a×(n-1)-a∴

a（n+1）=sn+（n+1),递推一项a(n)=s(n-1)+n两式相减a(n+1)-a(n)=a(n)+1,所以a(n+1)+1=2*(a(n)+1)a(n)+1成等比,推得a(n)+1=2^(n

an=Sn-Sn-1(n>=2)an=1/2a(n-1)-1/2a(n-2)=(1/2)a将a=1代入an不符,则该数列以分段的形式构成an=1(当n=1),an=1/2a(n>=2)

错项相减法

an+Sn=2n令n=1a1+S1=2=>a1=1又a（n-1）+S（n-1）=2（n-1）与上式作差an-a（n-1）+an=22an-a（n-1）=2an-2=（1/2）[a（n-1）-2]得证a

a(n+1)=3an+2a(n+1)+1=3(an+1)an=3^(n-1)*(a1+1)-1=2*3^(n-1)-1Sn=2(1+3+……+3^(n-1))-n=2*[(3^n-1)/(3-1)]-

题目条件应为：Sn=3an+2an=Sn-S(n-1)(n≥2)=3an-3a(n-1)(n≥2)=>an/a(n-1)=3/2.∴数列{an}成等比数列当n=1时,a1=3a1+2a1=-1.=>a

∵s[n]=n^2a[n]∴s[n+1]=(n+1)^2a[n+1]将上述两式相减,得：a[n+1]=(n+1)^2a[n+1]-n^2a[n](n^2+2n)a[n+1]=n^2a[n]即：a[n+

2=a(k)+a(n-k),2=a(k)+a(n+1-k).2=a(1)+a(n+1-1)=a(2)+a(n+1-2)=a(3)+a(n+1-3)=...s(n)=a(1)+a(2)+a(3)+...

(An)^2=2Sn-An=>(A(n-1))^2=2S(n-1)-A(n-1)=>(An)^2-(A(n-1))^2=2Sn-An-2S(n-1)+A(n-1)=>(An+A(n-1))*(An-A

/>n≥2时,Sn=n²×anS(n-1)=(n-1)²×a(n-1)an=Sn-S(n-1)=n²×an-(n-1)²×a(n-1)(n²-1)an

（1）3an=2Sn+n...①3an+1=2Sn+1+n+1...②②-①得：3an+1-3an=2an+1+1即an+1=3an+1==＞an+1+1/2=3(an+1/2)an+1+1/2/an