Question

已知数列an中,a1+3a2+5a3+……+(2n-1)an=(2n-3)2n+1 ,求an及Sn？

Answer 1

由题设a1+3a2+5a3+...+(2n-1)an=(2n-3)*2^(n+1)……①,将1式中的项数换为n-1.有：a1+3a2+5a3+...+(2n-3)an-1=(2n-5)*2^n……②,1式减去2式,得：(2n-1)an=(2n-3)*2^(n+1)-=(2n-5)*2^n=(2n-1)2^n,所以an=2^n,7,a1+3a2+5a3+……+(2n-1)an +(2n+1)*an+1 =[2(n+1)-3]×2^[(n+1)+1] 两式相减就得到 (2n+1)*an+1 = [2(n+1)-3]×2^[(n+1)+1] - (2n-3)×2^(n+1) a(n+1)=2^(n+1) an=2^n (n>=2) a1 ==(2*1-3)×2^(1+1)=-4 Sn=a1+a2+……+an=2+4+8+……+2^n -6=2^(n+1) -8,2,